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Structure, stability properties, and nonlinear dynamics of lateral modes of a broad areasemiconductor laser

Blaaberg, Søren

Publication date:2007

Document VersionEarly version, also known as pre-print

Link back to DTU Orbit

Citation (APA):Blaaberg, S. (2007). Structure, stability properties, and nonlinear dynamics of lateral modes of a broad areasemiconductor laser.

Structure, stability properties, andnonlinear dynamics of lateral modes of

a broad area semiconductor laser

Søren Blaaberg Jensen

Ph.D. thesis

COM·DTUNovember 2006

Structure, stability properties, and nonlinear dynamics oflateral modes of a broad area semiconductor laser

Søren Blaaberg Jensen

This thesis is submitted as part of the requirements for obtaining the Ph.D.degree from the Technical University of Denmark. The project was fundedby Risø National Laboratory.

Supervisors:

Bjarne Tromborg Department of Communications, Optics and MaterialsKarsten Rottwitt Department of Communications, Optics and MaterialsPaul Michael Petersen RisøNational Laboratory

Søren Blaaberg Jensen Kgs. Lyngby, November 6, 2006

i

Abstract

The subject of this thesis is a theoretical investigation of the nonlinear lateralmodes of broad area (BA) semiconductor lasers including studies of station-ary properties, of stability properties, and of the dynamics of a BA laser inan external cavity.

The most prominent characteristics of the output field of a BA laser aredue to lateral properties. A detailed investigation of stationary lateral fielddistributions is carried out and leads to the finding of a systematic structureof several categories of lateral nonlinear modes. In addition to the knowndefinite-parity modes, asymmetric modes are found, although the physicalsystem under investigation is symmetric. The structure and interrelation-ship between different modes are also seen in their tuning curves.

A stability analysis of the above mentioned stationary solutions must becarried out in order to evaluate their physical role. By means of a Green’sfunction method a small-signal analysis is carried out with emphasis on thestability properties. It is found that all regarded modes are unstable exceptfor the case of very low pump currents. The small-signal stability analysisexplains why BA lasers are generally found to have fluctuating output fields:at considerable pump currents there are no stable stationary solutions.

An existing external-cavity scheme including a spatial filter is imitatedtheoretically and it is found that one consequence of the external cavityis to dampen the lateral dynamics of the field, which in turn leads to aimprovement of the spatial coherence of the output. The near-field revealsthat the external-cavity scheme changes the lateral dynamics of the BA laserto a behavior more similar to a laser array.

ii

Resume

Titlen pa dette Ph.d.-projekt er “Struktur, stabilitetsegenskaber og ikke-lineær dynamik i bred-areal halvlederlasere”. Indholdet er en teoretisk un-dersøgelse af ikke-lineære laterale modes i bred-areal-lasere (BA-lasere). Un-dersøgelsen omfatter stationære egenskaber, stabilitetsegenskaber samt dy-namiske egenskaber af en BA-laser i en ekstern kavitet.

De mest tydelige karateristika ved en BA-lasers udgangsfelt skyldes lat-erale egenskaber. En detaljeret undersøgelse af stationære løsninger til detlaterale feltproblem udføres og leder til erkendelsen af en systematisk struk-tur i flere kategorier af laterale ikke-lineære modes. I tillæg til kendte modesmed bestemt paritet findes asymmetriske modes selvom det betragtede fy-siske system er symmetrisk. Strukturen og indbyrdes tilhørsforhold mellemstatinonære tilstande ses ogsa pa deres tuningskurver.

En stabilitetsanalyse af de ovenfor omtalte stationære tilstande udføresfor at evaluere deres fysiske betydning. Ved hjælp af en Green’s funktionmetode udføres en smasignalanalyse med hovedvægt pa stabilitetsegensk-aber. Det viser sig at alle betragtede stationære løsninger er ustabile bortsetfra ved meget lave pumpestrømme. Den lineære stabilitetsanalyse forklarerhvorfor BA-lasere generelt har fluktuerende udgangsfelter: Ved betragteligepumpestrømme er ingen stationære løsninger stabile.

Et eksisterende ekstern-kavitets-system der indbefatter et rumligt filterimiteres teoretisk og det bliver klart at en konsekvens af den eksterne kaviteter at dæmpe nær-feltes laterale dynamik, hvilket medfører en forbedret rum-lig kohærens. Nær-feltet afslører at den eksterne kavitet ændrer BA-laserenslaterale dynamik til mere at ligne dynamikken i et laser array.

iii

Acknowledgments

Many thanks to my main supervisor Bjarne Tromborg for sharing some ofhis wisdom with me and for his generous engagement in the project. It wasalways a pleasure spending time with Bjarne.Also thanks to Karsten Rottwitt for stepping in as main supervisor the lastfew months after Bjarne retired. I wish all the best to Søren Dyøe Agger inthe years to come. Colleagues at COM are acknowledged, especially a longstring of nice office mates. The last of whom was Niels Gregersen who hadto put up with my foot tapping.Special thanks go to Lone Bjørnstjerne for her kind help on many occasions.

At Risø I thank Paul Michael Petersen for funding and many nice peoplefor some good times.

iv

Contents

1 Introduction 1

2 Broad area semiconductor lasers 52.1 Broad area laser devices . . . . . . . . . . . . . . . . . . . . . 62.2 Spatiotemporal behavior . . . . . . . . . . . . . . . . . . . . . 82.3 Schemes aimed to increase coherence . . . . . . . . . . . . . . 9

2.3.1 On-chip schemes . . . . . . . . . . . . . . . . . . . . . 102.3.2 External cavity schemes . . . . . . . . . . . . . . . . . 11

2.4 Asymmetric external cavity laser . . . . . . . . . . . . . . . . 112.5 Theory and modeling of BA lasers . . . . . . . . . . . . . . . . 15

3 Theory of stationary lateral modes in Broad area lasers 193.1 Derivation of equations using the mean field approximation . . 203.2 A single nonlinear field equation . . . . . . . . . . . . . . . . . 27

3.2.1 Current spreading . . . . . . . . . . . . . . . . . . . . . 303.3 Numerical procedures for calculation of modes . . . . . . . . . 31

3.3.1 Solutions with definite parity . . . . . . . . . . . . . . 313.3.2 Asymmetric solutions . . . . . . . . . . . . . . . . . . . 323.3.3 Parameter values . . . . . . . . . . . . . . . . . . . . . 33

3.4 Calculated stationary solutions . . . . . . . . . . . . . . . . . 343.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4 Small-signal analysis using Green’s functions: Stability prop-erties and responses 534.1 Small signal analysis using the logarithmic field . . . . . . . . 554.2 Local Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . 634.3 Calculation of y . . . . . . . . . . . . . . . . . . . . . . . . . . 654.4 Global stability . . . . . . . . . . . . . . . . . . . . . . . . . . 66

v

4.5 Results of stability analysis . . . . . . . . . . . . . . . . . . . 674.5.1 Local stability of type I modes . . . . . . . . . . . . . . 674.5.2 Local stability of type II modes . . . . . . . . . . . . . 684.5.3 Global stability of type I modes . . . . . . . . . . . . . 694.5.4 Global stability of type II modes . . . . . . . . . . . . 734.5.5 Summary of stability properties . . . . . . . . . . . . . 74

4.6 Linewidth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754.7 Static Frequency tuning . . . . . . . . . . . . . . . . . . . . . 764.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

5 Time-domain calculations 815.1 Time-domain equations . . . . . . . . . . . . . . . . . . . . . . 825.2 External feedback without filtering . . . . . . . . . . . . . . . 845.3 Spatially filtered feedback . . . . . . . . . . . . . . . . . . . . 85

5.3.1 Single stripe mirror . . . . . . . . . . . . . . . . . . . . 875.4 Hopscotch method . . . . . . . . . . . . . . . . . . . . . . . . 895.5 The problem with adiabatic elimination . . . . . . . . . . . . . 895.6 Time-domain results . . . . . . . . . . . . . . . . . . . . . . . 91

5.6.1 Freely running laser . . . . . . . . . . . . . . . . . . . . 925.6.2 Asymmetric external cavity laser . . . . . . . . . . . . 95

5.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

6 Modal expansion of lateral modes 1076.1 Expansions of the lateral field and the carrier density . . . . . 1076.2 Linear gain guided modes . . . . . . . . . . . . . . . . . . . . 1106.3 Results of mode expansion . . . . . . . . . . . . . . . . . . . . 112

6.3.1 Comparison of modal expansion with scattering-potentialmethod . . . . . . . . . . . . . . . . . . . . . . . . . . 112

6.3.2 Perturbation interpretation of low-current nonlinear modes1136.3.3 Identification of most important expansion terms and

the influence of carrier diffusion . . . . . . . . . . . . . 1156.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

7 Summary 119

A Energy density and power 123

B Boundary conditions 125

vi

C The stability parameter σ 129

D The Diffusion matrix D(x, s) 133

E The hopscotch method 135

F Modal expansion -method of solution 141

G Papers and presentations 143

vii

Chapter 1

Introduction

Broad area (BA) semiconductor lasers are edge-emitting semiconductor lasersusually designed as high-power laser devices intended to emit as much poweras possible while at the same time having a reasonably long lifetime. Theyare wide-aperture Fabry-Perot lasers which, in their simplest form, only offerguiding of light by means of index guiding in one transverse direction; inthe second much broader transverse direction (the lateral direction) the lightis purely gain guided. An injected current inverts the semiconductor gainmaterial over a limited lateral region where light is amplified. The broadnessof the lateral region pumped by the injected is motivated by a combined urgefor high output power along with the necessity of lowering the intensity oflight to avoid catastrophic optical damage.

BA lasers, being Fabry-Perot lasers, are longitudinally multi-moded. How-ever, the main interest both from a theoretical and an application point ofview has traditionally been directed towards the lateral behaviour. ModernBA lasers usually have emitter-widths of 100 µm or wider. The geometryof BA lasers gives a single-mode behavior in the index guided transverse di-rection while the behaviour in the lateral gain guided region typically givesrise to a heavily laterally multi-moded behaviour yielding complex variationsin space and time often termened filamentation giving inherently poor co-herence properties. Consequently, various schemes aimed at improving thecoherence of the output of BA lasers have been suggested.

While the incoherent output is of inconvenience for applications demand-ing high power, BA lasers form a laboratory for the study of nonlinear phe-nomena. The work presented in this thesis is theoretical. The purpose of

1

the work is to obtain a better understanding of the lateral properties of BAlasers. While the dynamics of BA lasers has been studied quite a lot, thestationary mode structure has been studied less systematicly. We perform anin-depth investigation of the stationary lateral modes in a BA Laser. Afterfinding stationary solutions in the lateral nonlinear system, we go through asmall-signal analysis. In time-domain calculations we try to reproduce exper-imental behavior of a set-up involving a BA laser in an asymmetric externalcavity acting as a spatial filter. The purpose of the cavity is to improve thespatial coherence of the BA laser. The contents of the remaining chapters isoutlined below.

In Chapter 2 we present the regarded BA-laser device. It is a genericstructure as our aim is to do a general investigation rather than to view aparticular design. Motivated by the need for lasers with improved coherenceproperties, an existing external-cavity scheme aimed at improving the spatialcoherence is described. We study this secheme in a later chapter. We lastlyaim to give a very brief overlook of the theoretical treatment of BA lasers,while mentioning the approaches we pursue in the subsequent chapters.

In Chapter 3 we first derive the main equations governing the lateralfield distribution and carrier-density distribution using mean-field theory.Here mean-field theory yields performing an average over the longitudinaldirection of the laser. We then combine these two equations to form a single,nonlinear field equation with the aim of finding stationary solutions. Withappropriate boundary conditions for lateral gain guided modes, stationarysolutions are then found. We find that the variety and structure of lateralstationary solutions and the way their frequencies vary with pump currentto be much richer than what has previously been shown [1]. We find asym-metric modes in the symmetric laser structure and other modes which canonly exist due to the nonlinear nature of the gain material. The modes turnout to be related in a systematic structure that we have found in their tun-ing curves, i.e. curves showing the stationary frequency of the modes versuspump current, and in their field distributions.

Along with the results of Chapter 3 yielding stationary solutions , Chap-ter 4 is of a theoretical nature. In this chapter we perform a small-signalanalysis of the nonlinear stationary solutions obtained in Chapter 3 using aGreen’s function approach. In particular we investigate the stability proper-

2

ties of selected modes. The analysis shows that all investigated lateral modesare unstable with the exception of the two lowest order modes at very lowpump currents. This fundamental result is in correspondance with streak-camera measurements [2] and large-signal theory [3][4] which tell that BAlasers are never in a steady state when the applied current in considerable.

Chapter 5 contains time-domain calculations based on the numericalmethod named hopscotch. A solitary BA laser is compared to a BA laserin an asymmetric external cavity. The external cavity improves the spatialcoherence of the lsaser. We obtain a good qualitative agreement with mea-surements found in the literature. Our calculations show that the asymmetricexternal cavity laser operates in a fluctuating state.

In Chapter 6 we present a method to calculate lateral modes in a BAlaser via a mode expansion. With an expansion in linear gain guided lateralmodes, it becomes possible to recognize the most significant perturbations ofthe field at low currents.

Chapter 7 gives a short summary of results.

3

4

Chapter 2

Broad area semiconductorlasers

The geometry of BA-laser devices with their wide apertures allow for high-power output when pumped at high currents. In fact the geometry alsomakes BA lasers suitable for scientific purposes as testbeds for gain ma-terials [5] or for experiments on spatially nondegenerate four-wave mixing.Injecting a pump- and a probe beams at different angels with a frequencydetuning makes it possible to measure ambipolar carrier diffusion coefficientsand carrier lifetimes [6]. Their usefulnes as lasers, however, is also limitedby the geometry since it allows the lateral field distribution to vary in timeand space in a complex manner that ruins the coherence of the output beam.The strongly nonlinear behavior due to the light-semiconductor interactionalso gives a range of interesting phenomena to be studied. Moreover, thehunt for methods to improve the coherence of the output of BA lasers hasbeen ongoing for at least two decades.

Now, we desribe the basics of BA lasers, and qualitatively discuss theirspatiotemporal behavoir which is often described through the process of fil-amentation. We then, after briefly reviewing methods to improve coherenceproperties of BA lasers, describe an asymmetric external cavity BA laser thatwe investigate in Chapter 5. Lastly in this chapter, we discuss some of thedifferent paths that one can choose, including those that we choose, whenone studies BA lasers theoretically.

5

2.1 Broad area laser devices

The name “BA laser” originates from the laser geometry. When increasingthe current in a semiconductor laser high above its threshold, the intensityof light eventually surpasses a threshold where catastrophic optical damageis done to the laser facets. At the same time many applications demandhigh-power output. In order to keep the intensity of light below the damagethreshold of the laser facets and at the same time obtain high output power,the laser structure is made wider in the lateral direction.

x=0

r2r1

x=x0

x=−x 0

Aly’

Alx’

Alz’

z=l

Output

Metal contact

y

x

z

z=0

Ga

Ga1−x’

1−z’Ga

1−y’As (p−type)

As

As (n−type)

Figure 2.1: Generic structure of a broad area laser. The active layer(Alx′Ga1−x′As) is sandwiched between two cladding layers. The coordinatesystem represents the lateral (x), the transverse (y), and the longitudinal (z)axes, respectively. The origin is centered in the middle of the waveguide atthe back facet. Here it is dispalced for clarity.

BA lasers are edge-emitting lasers. Figure 2.1 illustrates a BA laser in itssimplest form. Three semiconductor layers form a p-i-n junction. The toplayer is a p-type doped cladding layer. The middle layer, an intrinsic corelayer, is the active layer where light may be amplified in case the materialis inverted. The bottom layer is an n-type doped cladding layer. On top ofthe structure sits a metal contact. Not shown in the figure is the substrateon which the n-type layer is grown. Under the substrate a second metalcontact is deposited. In the calculations presented in this thesis we haveassumed the material composition of the BA laser to be in the AlGaAs

6

material system but this is not essential for the theoretical analysis. FromFigure 2.1, it is evident that there is no lateral (in the x-direction) variation inthe material composition and therefore no lateral variation in the refractiveindex present in the system. The BA laser regarded here is thus purelygain guided. For a purely gain guided BA laser the optical field is limitedin the lateral direction only by the extend of the region having appreciablecurrent pumping. BA lasers with weak lateral index guiding [7] and morecomplicated and refined layer structures [8] than the simple one in Figure2.1 have been realized. However, the work presented in this thesis is purelytheoretical and has no relation to a specific device wherefore we focus ona generic BA-laser structure. This generic structure is thus a wide doubleheterostructure with a wide top metal contact. The double heterostructureserves two important purposes. Since the bandgaps of the p- and n-layers arehigher than the one of the intrinsic layer, carriers are confined in the intrinsiclayer where they are to recombine preferably under stimulated emission. Inaddition, the intrinsic layer also confines light as its refractive index is higherthan the two outer layers. The double heterostructure is therefore also aslab waveguide responsible for the transverse (y-direction) guiding of light.The core layer is made sufficiently thin so that only one transverse mode issupported. Most if not all commercially available BA lasers have one or morequantum wells serving as the gain material. When quantum wells constitutethe active region of the laser a separate-confinement heterostructure mustcarry the burden of waveguiding. BA lasers with quantum dot gain materialhave also been reported [9].

For a gain guided single-stripe AlGaAs laser supporting only one lateralmode, the width of the top metal contact (the current stripe) is typically 3to 5 µm. The top metal contact of a BA laser can be said to be one or twoorders of magnitude wider than that of a singe-lateral mode laser. BA laserswith current stripes as wide as 1000 µm have been reported [10]. In thisthesis we regard a width of w = 2x0 = 200 µm. When increasing the widthto several tens of microns, one allows for lateral multimode operation. In factwhen increasing the width of the current stripe to obtain a higher maximunoutput power, the high-power performance is limited by spatially localizedbursts (filaments) of high intensity, which in itself lowers the threshold forcatastrophical optical damage of the output facet [11]. Nevertheless, thehighest output power achievable from a semiconductor laser increases withincreasing area of the output facets. Therefore BA lasers remain poplar forhigh-power applications. The laser mirrors (facets) with amplitude reflectiv-

7

ities r1 and r2 are obtained by cleaving the crystal in planes perpendicularto the grown layers. The facets may in addition be coated to modify theirreflectivities.

The p-i-n junction is forward-biased to obtain inversion. When the diodeis forward biased electrons and holes are injected into the intrinsic layerwhere they may recombine or continue to the layer opposite to the one fromwhich they were injected. As the forward bias is increased the quasi-Fermilevels of the electrons and the holes, will increase and decrease, respectively.When they are separated by the bandgap energy, the material is inverted.In semiconductor laser modeling it is often assumed that one is not too faraway from thermal equilibrium and that all electron-hole transitions takeplace between the extrema of one conduction band and one valence band,i.e. at zero wave number. In this thesis we will employ this approach.

2.2 Spatiotemporal behavior

In most experimental work on high-power lasers, slow detection methods av-erage out any fast variations in time giving only a static spatial variation inthe intensity distribution to read out. Such measurements are unlikely togive a full understanding of the physical mode of operation of a particularlaser. Fischer et al. [2] measured the spatio-temporal dynamics of the outputfield of a BA laser on a picosecond time scale using a streak-camera. Thenear-field was seen to consist of rapidly changing irregular lateral patternsof light intensity. A BA laser pumped at a high current never finds a steadystate. The dynamics following the initial relaxation oscillations may be di-vided into two domains of filaments [12] where a filament is a small regionin the active region of relatively high intensity. Firstly, “static” filamenta-tion where regions of the near-field of high respectively low intensity retaintheir individual lateral positions. The field of a BA laser is not static, how-ever, even at moderate (moderate not being high) pump currents. Thus the“static” filaments are turned on and off on a time scale of the order of 100 ps.The reason for this is “dynamic filamentation”, the second domain, whichmeans that the filaments tend to move laterally as a function of time. Thisbehavior manifests itself as zig-zag patters in the temporal evolution of thenear-field (we shall see an example of this in Chapter 5). If a filament origi-nating on one edge of the active region migrates all the way to the oppositeedge it would take roughly between 200 to 500 ps for the inspected device

8

of [2]. The device was a GaAs/AlGaAs device of width 100 µm pumped attwo times threshold. Static theories of filamentation have been also been putforth [13]. In fact one may interpret stationary field distributions obtainedby solving an appropriate set of model equations, i.e. a nonlinear spatialmode in the laser, as static filamentation [1][14]. In Chapter 3 we find sev-eral different types of nonlinear modes revealing a rich variety of stationary,spatial shapes.

The irregular output of BA lasers has its origin in the process of dynamicfilamentation. In BA lasers with no passive, lateral index guiding, the lateralfield distribution is constrained laterally only by the finite width of the lat-eral current distribution. Antiguiding in semiconductors is the phenomenonwhere regions with relatively high carrier density or inversion (and thereforea relatively high gain) implies a relatively low refractive index. Oppositely forregions with relatively low carrier density (and therefore a relatively low gain)which have a relatively high refractive index. The filamentation process in-volves antiguiding that causes self-focusing, diffraction, and local differencesin gain: Consider a local burst of high intensity (a filament). The carrierdensity is locally depleted causing locally low gain and due to the antigu-iding effect locally high refractive index compared to the surrounding areawhere the gain is relatively high and hence the refractive index relativelylow. The filament can persist due to the index guide which has been formed.Eventually, however, the gain in its neighboring area, where the intensity oflight is low, rises sufficiently high above the threshold level, due to the pumpcurrent, so that the filament moves laterally and is amplified and the processcan start over. With many such filaments interacting nonlinearly the overallresult is an apparently chaotic behavior in time and space. We will showexamples of this behavior in Chapter 5. The above description of dynamicfilamentation relies on local depletion of the carrier density. At the two edgesof the pumped region the carrier density is typically relatively high due to arelatively low intensity of light indicating that the total field creates a globalwaveguide through gain-guiding and anti-guiding. By global we mean on thelength scale of the width of the metal contact w = 2x0.

2.3 Schemes aimed to increase coherence

Because BA lasers can deliver high-power output but have weak coherenceproperties, there has been an urge to improve the latter. It seems that the

9

largest effort has been to improve the spatial coherence in order to be ableto focus the output beam e.g. into an optical fiber. To optimize the spatialcoherence, then, means to obtain an output resembling a Gaussian beam toas high an extend as possible. Here we briefly mention a few attempts totame the beast, after which we will describe the specific external-cavity (EC)setup which we regard in Chapter 5. We divide the schemes into on-chipschemes where advanced semiconductor technology has been used in orderto tailor a laser cavity to give a desired stable single-mode output or oftena, more realisticly speaking, partly stabilized output, and then EC schemeswhere there is a finite delay between the output of the BA laser and the fieldthat is fed back to the laser.

2.3.1 On-chip schemes

The α-distributed-feedback laser [15][16] is essentially a BA laser with a singleintra-cavity angled grating whose fringes make up a substantial angle withthe lazer axis (z-axis). The current stripe is angled parallel to the grating.The grating filters the intra-cavity field spatially and spectrally giving single-mode operation both spatially and spectrally. The output-field is close tobeing Gaussian and filamentation is well suppressed.

Another on-chip scheme has been demonstrated with broad area lasershaving an intracavity spatial phase controller yielding a nearly diffraction-limited single-lobed far-field [17]. In the demonstrated lateral-multi-segment-device, a spatial phase controller could generate an asymmetric lateral varia-tion in the longitudinal optical path length. Pulsed powers of 300 mW wereachieved. The authors suggest that a tilted end facet (∼ 1 degree) shouldhave an equivalent effect. We mention this device as it has a built-in lateralasymmetry. The EC laser that we study in Chapter 5 is also asymmetric.The device had a single-lobed far-field off the laser axis. Semiconductor-laserarrays are devices related to BA lasers, which provide an effective methodto suppress the filamentation in a wide-aperture laser. Also, a strong com-petitor to the BA laser as a high-power device are tapered lasers. Praci-tally diffraction-limited tapered lasers with multi-Watt output emerged inthe mid-nineties [18].

10

2.3.2 External cavity schemes

External cavities offer the combination of an delayed optical feedback andoptionally a filtering. For twin-stripe lasers the effect of the delay in an ex-ternal cavity can in specific cases be seen to stabilize what was a chaoticoutput intensity for the solitary laser to a periodically varying output [19].For BA lasers with their broad spatial spectrum, some means of spatial filter-ing is probably necessary to increase the spatial coherence and/or stabilizethe dynamic filamentation. Most likely one cannot have the former withoutthe latter at high pump currents. In [20] a spatially filtered feedback bymeans of a tilted plane mirror was applied and streak-camera measurementsshow that the filter is able to suppress the dynamic filamentation rather well.Another way to achieve spatial filtering is to use an external reflector witha finite radius of curvature. Such external cavities have produced operationcausing a single-lobed far-field of the laser [21] in agreement with theoreticalpredictions [22] [23]. Phase conjugate feedback without spatial filtering hasproduced operation in a single longitudinal mode [24]. Even operation ina true single longitudinal and lateral mode has been achieved albeit at lowcurrents using photorefractive feedback [25]. It should be noted that a planeconventional mirror (without any spatial filtering) has been shown not tohave any stabilizing effect on the output of a BA laser, nor to bring it to lasein a single longitudinal or lateral mode [24].

2.4 Asymmetric external cavity laser

It has been shown experimentally in [26][27] and through modeling [28] thatinjection set-ups with a single-mode laser acting as a master oscillator andthe BA laser amplifier as the slave, that the best spatial coherence of theoutput of the slave is achieved when the angle of incidence of the masterbeam on the front facet of the BA laser is off the laser axis of the BA laser,i.e. the slave. On the contrary, normal incidence (θ = 0) of the masterbeam causes filamentation in the near-field and a far-field distributed over alarge range of angles. It thus appears that locking the fundamental spatial(lateral) mode is difficult to achieve. Considering an external-cavity schemein which the intention is to force a wide aperture laser to ideally oscillatein a single lateral mode, it makes good sense to enhance lateral modes ofthe laser that emit light away from the laser axis in the far-field. For this

11

purpose an asymmetric external cavity (AEC) laser was introduced in 1987[29]. The original work was done using a wide semiconductor laser array butthe behavior of the system holds similar to the case when a BA laser is usedas the active part of the system.

x=0

x=−x 0

x=x 0

r2r1

f f

r3

z=lz=0

yz

x

Output

FF

Reflector

y−Axis Collimator

Figure 2.2: Top view of AEC laser. BA laser emits ligth through right facet.Lens of focal length f Fourier transforms the field from the x-domain to thekx-domain. The far-field (FF) or the Fourier plane is thus at z = l + 2f ,where the reflector with amplitude reflectivity r3 acts as a spatial filter (dueto its small lateral extend). The part of the field that is reflected due to r3

is again Fourier transformed to the x-domain. The y-axis collimator (a lens,e.g. a cylindrical lens, of short focal length) collimates the field in the quicklydiverging y-direction.

A version of the AEC laser made as simple as possible is presented inFigure 2.2. It is similar to [10]. It consists of a wide aperture laser, withinthis thesis a BA laser, two lenses, and a reflector (a stripe mirror) withamplitude reflectivity r3. The field emitted from the right facet of the BAlaser with amplitude reflectivity r2 propagates through the lenses before partof it is reflected by the external stripe-mirror width. A part of the reflectedfield returns to the right facet through the lenses. Let us describe the systemin more detail: Usually one assumes that the output (scalar) field of thesolitary BA laser can be written as a product E+(x, z = l)φ(y). The y-dependent part of the field is a single-moded and nearly Gaussian profilewith negligible phase curvature because of the index guiding of the double

12

heterostructure in the y-direction [30]. The lens labeled “y-axis collimator”,is placed immediately front of the output facet. It can be an e.g. cylindricallens. It collimates the field along the y-axis and the y-dependence can bedisregarded in the external cavity. The collimation is important since thefield is quickly divergent in the y-direction. In reality parts of the reflectedfield will not make back to the right facet because e.g. undesired scatteringat the external mirror or misalignments. These losses can from a theoreticalpoint of view be included in the reflectivity r3. The second lens is placedat z = l + f where f is the focal length of this lens. This means that thethis lens performs a spatial Fourier transform on the output field so thatat z = l + 2f (the Fourier plane) one obtains the spatial spectrum (the kx-domain) or the far-field of the output field. At the Fourier-plane the fieldis filtered and reflected by the external stripe-mirror and due to the returnto right facet at z = l through the lens, the field is transformed back to thex-domain. Speaking in broader terms any type of filter can be placed in theFourier-plane be it symmetric or asymmetric. Also, the lens could also bedisplaced from z = l + f to obtain a focused feedback [31].

It should be noted that in some reported set-ups, e.g. [10] [32], an aper-ture was inserted in the output arm as an additional spatial filter. The aimof the aperture is to separate unwanted side lobes from the dominant single-lobe in the far-field (we call the dominant lobe “the single-lobe”). Thereforereported single-lobe output at very high pump currents may have undergonesuch additional spatial filtering. Figures 2.3 and 2.4 adopted from [33] exem-plifies measured time-averaged near- and far-fields of a BA laser. The figurescompare the output of the solitary BA laser and the output when the AEC isadded. In Figure 2.4 the effect in the far-field is evident: the far-field of thesolitary laser is a blurred shape spread over a wide span of angles, typically2 to 6 degrees depending on the device and pump current. When the AECis added and optimized one sees a dominant single-lobe. The single-lobe isseen to be to the left of the optical axis. In this case the stripe mirror isplaced to the right of the optical axis. For a 200 µm-wide BA AlGAas-devicepumped at 2 times threshold the single-lobe of the AEC laser can be locatedaround 2 degrees off the optical axis in the far-field [34]. In order to obtain ameasurement as the one in Figure 2.4 showing the entire far-field, one mustinsert a beam splitter just before the external mirror. If one measures theoutput after the external mirror, one will only (or mainly) see the single-lobe.The near-field in Figure 2.3 shows that the effect of the external mirror is totilt the near-field. One can say that the near-field tilts in the same direction

13

as the far-field.AEC lasers with external cavities including a grating as the reflector [35]

and a grating as a reflector in combination with a Fabry-Perot etalon [36]have been shown to considerably narrow the emission spectrum while alsoimproving the spatial coherence. The spectra were narrowed from around 1nm less than 0.1 nm around 810 nm. For AEC lasers with a conventionalstripe-mirror, on the other hand, the spectral width of the freely runningBA lasers is not reduced significantly [37]. From a theoretical point of view,it is of interest whether an AEC laser with a mirror reflector operates in asingle, stable, lateral mode or in some time dependent state. Our calculationpresented in Chapter 5 implies that the latter is the case.

In high power laser technology, often the spatial coherence is of greaterconcern than the temporal coherence. A measure of spatial coherence oftenused in connection with high-power lasers is the M 2-factor. It expresses thesimilarity between a regarded output field of a laser with a Gaussian beam.The Gaussian has same width as the regarded field at its waist [38]. The M 2-factor appears to make most sense at values that are not vastly greater than1. The beam quality of the single lobe in the far-field of a AEC laser maybe measured using M 2. When pumped far above threshold, the M 2 of AEClasers degrades for increasing pump currents. Hence, a compromise betweenhigh output power and low M 2 must be made in high-power applications.We will not discuss the M 2-factor further.

Figure 2.3: Measured near-field of solitary laser (left) and AEC laser (right).The device was a BA laser with a 200 µm-wide current stripe running at 810nm. Adopted from [33].

14

Figure 2.4: Measured far-field of solitary laser (full line) and AEC laser(dotted line). Same case as Figure 2.3. Note that the abscissa-unit is lengthand not angle as it is common for the far-field. The optical axis (θ = 0) isaround 2700 µm. The stripe mirror is located around 3500 µm, i.e. oppositethe dominant single-lobe. Adopted from [33].

2.5 Theory and modeling of BA lasers

From approximately 1990 until today, mainly two paths have been followedto model BA lasers. Those are the beam propagation method (BPM) andtime-domain calculations by integration of time dependent partial differentialequations (PDEs).

BPM is a method to find stationary field distributions by propagating afield back and forth in the laser cavity, while considering the coupling betweenthe intensity of the field and the semiconductor, until a steady state has beenobtained. Its main quality is that it gives 2-dimensional laser modeling (asopposed to 1-dimensional) at low computational cost. We have implementedBPM both with and without the external cavity of the AEC laser. As it wasalso found in [1] we have found it problematic for the method to find steadystates except for at very low pump currents, especially when including an ex-ternal cavity in the system. Supposedly this problem is due to the differencein time scales of the optical and carrier rate equations [1]. Moreover, the

15

steady state which BPM may find depends on the trial field that is initiallylaunched into the system. Since the field is propagated back and forth in thecavity the effect of filamentation interferes with the aim of finding stationarysolution. An amusing example of the effect of filamentation on BPM can befound in [30] where the linewidth enhancement factor α was simply set tozero in order to get stationary results for a BA laser in an external cavity.With α = 0 there is no self-focusing, and hence no filamentation. We havenot found BPM suited for our purposes. When modeling BA amplifiers (notlasers) [39], in conjunction with injection locking of BA lasers [28], or forindex-guided devices such as tapered lasers [40], BPM may work well.

When recognizing the very complex fluctuations in time and space of BAlasers and desiring to be able to compare theory with experiment at consider-able pump currents it becomes advantageous to use a time-domain method.In fact one cannot expect stationary solutions to be found in measurementsat high currents. While treating the field of the laser classically, differentlevels of describing the semiconductor gain material have been regarded. Aphenomenological description using linear gain and the α-parameter is an“obvious” possibility [41]. However, as we find in Chapter 5 the phenomeno-logical model causes some problems in a spatially extended system where thediffraction of the field has to be included. Fortunately, one can modify theequations slightly to overcome the problem. Microscopic models using thesemiconductor Maxwell-Bloch equations [3][4] have been the other extremeat least for bulk BA lasers. In most cases known to us the numerical ap-paratus upon which the time-domain approaches rely, regardless of the levelof describing the semiconductor, is the hopscotch method, a method to inte-grate parabolic PDEs. Implementations including 2 and 1 spatial dimensionshave been presented. In Chapter 5 we use a 1-dimensional implementationof the hopscotch method with a phenomenological description of the semi-conductor. This has been computationally highly advantageous since all ournumerical calculations have been performed on a laptop computer. To workin one spatial dimension we use a mean-field approxiamtion when derivingequations in Chapter 3, implying that we average over the longitudinal di-rection z.

Of course one can choose other paths than the two described above.Within other types of lasers such as lateral single-mode EC lasers and dis-tributed feedback (DFB) lasers there has been a great tradition of findingstationary solutions of the given system and then investigating their small

16

signal properties. For example for a lateral single-mode EC laser, under cer-tain feedback conditions, the stationary solutions of the laser may be locatedon an ellipse in the (frequency, threshold-gain)-plane [42]. Half of the station-ary solutions are found to be unstable when subject to a stability analysis,and the laser may choose to lase in only one of the solutions on the ellipse orperhaps the lasers chooses a chaotic state, but this does not mean that theexistence of the ellipse is uninteresting. Based on these considerations wefind stationary lateral modes in a BA laser in Chapter 3 and perform a smallsignal analysis of some of the found modes in Chapter 4. Since a BA laseris known to operate in a fluctuating possibly chaotic state when driven atconsiderable currents, our analysis is of a theoretical character. The methodwe use to obtain stationary solutions resembles finding bound states in ascattering potential. Again, there are several lateral modes in a BA laserfor one pump current. With BPM this multitude of lateral modes would beextremely difficult to come about since the solution to which BPM settle,is dependent on a x-dependent trial field injected from one end of the laserwhen initiating the iteration.

Further, in order to disassemble the nonlinear effects perturbing the fieldnear threshold we introduce a modal-expansion technique that allows for aneasy interpretation in Chapter 6.

17

18

Chapter 3

Theory of stationary lateralmodes in Broad area lasers

In this chapter we first formulate the equations which form the basis of theresults presented in this thesis. The derived equations describe the lateralfield distribution and the lateral carrier density distribution of a BA laser.To study the lateral mode structure in detail we have chosen to reduce the,in principle, 3+1 dimensional problem to a 1+1 dimensional problem. Wethen combine the field and carrier density to a single, nonlinear equation forstationary solutions following Lang et al. [1].

Secondly, we calculate stationary lasing solutions. Here, a stationary so-lution is the combination of an oscillation frequency ωs, a stationary lateralfield distribution Es(x), and a stationary lateral carrier-density distributionNs(x). We have found a wide variety of modes in addition to known gainguided modes. It turns out that modes with asymmetric field distributionsexist despite the symmetric lateral structure under investigation. Further-more the stationary solutions yield a beautiful pattern of tuning curves whichpossess a systematic structure in their interrelationship and bifurcation be-havior. Tuning curves are curves in the current-frequency plane.

As discussed in Chapter 2 one cannot, in general, expect that calculatedstationary modes will agree with measurements. Stationary solutions maynot be stable, and one must at least investigate the stability properties ofgiven modes before discussing their conceivable role in an experiment. Weinvestigate stability properties of the stationary solutions in Chapter 4.

19

3.1 Derivation of equations using the mean

field approximation

We now derive a set of equations for the field distribution and the carrierdensity. We choose to employ a mean field approximation which impliesaveraging over the longitudinal direction of the laser. We assume that thelaser operates in a TE mode. The scalar wave equation in the time-domainis given as [43][44]

∇2E − σ

ε0c2∂

∂tE − 1

c2∂2

∂t2E =

1

ε0c2∂2

∂t2(P + p), (3.1)

where ω is the angular frequency. The real scalar electric field E (x, y, z, t)induces the polarization field P(x, y, z, t). The material losses are includedin the conductivity σ. Spontaneous emission is included in the term p. c isthe speed of light in vacuum and ε0 is the vacuum permittivity. The Fouriertransforms are defined as

Eω(x, y, z) =

∫ ∞

−∞E (x, y, z, t)e−jωt (3.2)

E (x, y, z, t) =1

2π

∫ ∞

−∞Eω(x, y, z)ejωt. (3.3)

In the frequency domain Eq. (3.1) becomes

∇2Eω − jωσ

ε0c2Eω +

ω2

c2Eω = − ω2

ε0c2(Pω + pω). (3.4)

For single-mode semiconductor lasers it has been a vast success to assumethat the polarization relaxes in time scales much faster than the time scales ofthe other variables in the problem, the field and the distribution functions ofthe charge carriers. If one adiabatically eliminates the polarization variablein the semiconductor Maxwell-Bloch equations (see [45] for the case of asemiconductor BA laser) then

Pω = ε0χω(x, y, z)Eω(x, y, z) (3.5)

where the susceptibility χω(x, y, z) is related to the permittivity εω(x, y, z)through

εω(x, y, z) = 1 + χω(x, y, z) − jσ

ε0ω. (3.6)

20

Using Eqs. (3.5) and (3.6) in (3.4) yields

[∇2 +

ω2

c2εω(x, y, z)

]Eω(x, y, z) = Fω(x, y, z), (3.7)

with

Fω(x, y, z) = − ω2

ε0c2pω(x, y, z). (3.8)

We now show how the problem of solving the 3+1 dimensional scalar waveequation can be reduced to solving a 1+1 dimensional wave equation byapplying a weighted mean field approximation. We assume that the electricfield in the BA-laser waveguide in the frequency domain is of the form

E (x, y, z, ω) = E+ω (x, z)φ(y)e−jβz + E−

ω (x, z)φ(y)ejβz. (3.9)

E+ω (x, z) and E−

ω (x, z) are field envelopes describing forward and backwardtraveling waves in the longitudinal direction. They are assumed to be slowlyvarying functions of z. We intend to study lateral modes for a given longi-tudinal mode. The propagation constant β is therefore chosen to satisfy thelongitudinal oscillation condition for a Fabry-Perot laser

r1r2 exp(−2jβl) = 1 (3.10)

where r1 and r2 are the left and right facet reflectivities (see Figure 2.1). Thefunction φ(y) describes the transverse field distribution and is taken to benormalized to unity, i.e

∫φ(y)φ∗(y)dy = 1. Inserting (3.9) in the scalar wave

equation (3.7) and neglecting the second order z-derivatives leads to

φ∂2

∂x2E±

ω +E±ω

∂2

∂y2φ∓2jβφ

∂

∂zE±

ω −β2E±ω φ+k2

0εω(x, y, z)E±ω φ = F±

ω , (3.11)

where k0 = ω/c is the vacuum wavenumber. By standard procedure weseparate into a transverse field equation

∂2

∂y2φ+ k2

0εω(x, y, z)φ = k2effφ (3.12)

and the in-plane field equations for E±ω (x, z)

∂2

∂x2E±

ω ∓ 2jβ∂

∂zE±

ω + (k2eff − β2)E±

ω = f±ω , (3.13)

21

where

f±ω (x, z) =

∫ ∞

−∞Fω(x, y, x)φ∗(y)dye±jβz . (3.14)

The eigenvalue equation (3.12) determines the fundamental transverse modeφ(y) and the corresponding effective wave number keff(x, z, ω). We ignorethe weak dependence of φ on x. The effective wavenumber is related to thecomplex effective refractive index by keff(x, z) = neff (x, z)(2π/λr) where λr

is a reference wavelength [43] [46]. neff(x, z) can be obtained by treatingthe loss and pump dependent part of εω(x, y, z) in (3.12) using first orderperturbation theory yielding a real pump independent effective index nr anda confinement factor

Γ =

∫active layer

|φ(y)|2dy∫∞−∞ |φ(y)|2dy . (3.15)

In (3.15) it has been assumed that the internal loss in the cladding layersare the same as in the core layer. The mean field approximation deals withaverages over the z-coordinate; an averaged variable is denoted by putting abar over the variable. Thus

E±ω (x) =

1

l

∫ l

0

E±ω (x, z)dz . (3.16)

The longitudinal average of (3.13) yields the equations

∂2

∂x2E±

ω ∓ 2jβ

l(E±

ω (x, l) − E±ω (x, 0)) + (k2

eff − β2)E±ω = f±

ω (3.17)

where we have assumed that [47]

k2effE

±ω = k2

effE±ω . (3.18)

The envelope fields E±ω obey the boundary conditions

E+ω (x, 0) = r1E

−ω (x, 0) (3.19)

E−ω (x, l) = r2e

−j2βlE+ω (x, l) (3.20)

at the two end facets. With β satisfying (3.10) we find that (3.17), (3.19)and (3.20) lead to the equation

∂2

∂x2Eω + (k2 − β2)Eω = fω (3.21)

22

for the weighted field and noise functions Eω(x) and fω(x) given by

Eω =1√2r1

(E+ω + r1E−

ω ) (3.22)

fω =1√2r1

(f+ω + r1f−

ω ) . (3.23)

Also we have defined k by k2 ≡ k2eff .

The frequency domain equation (3.21) can be transformed to a time do-main field equation for the complex field envelope E(x, t) defined by

E(x, t)ejωst =1

2π

∫ ∞

0

Eω(x)ejωtdω, (3.24)

where ωs is the optical frequency of the lateral mode under consideration. IfE±

ω are independent of z, the average photon density in the active layer isgiven as

S(x) = B|E(x, t)|2 (3.25)

with the constant of proportionality [48]

B =2ε0nrng

~ωhK (3.26)

where K is the longitudinal Peterman-factor [49]

K =(r1 + r2)(r1r2 − 1)

2r1r2ln(r1r2), (3.27)

and where we have used that the confinement factor Γ is approximately equalto h|φ(0)|2 with h being the thickness of the active layer. Details are givenin Appendix A. For most practical cases the factor K is close to one. nr isthe real passive part of the effective index and ng is the group index. We willassume that (3.26) is a useful approximation even when longitudinal spatialhole burning makes the field envelope E±

ω dependent on z.Solving for β in Eq. (3.10) yields

β =πpl

l+ j

αm

2(3.28)

where pl is an integer denoting the longitudinal mode number and αm is thedistributed mirror loss

αm = −1

lln(r1r2). (3.29)

23

As we regard only one longitudinal mode we set the real part of (3.28) equalto a reference wave number kr

kr =ωr

cnr, (3.30)

where ωr = 2πc/λr is a corresponding reference frequency. When there is nolateral, passive index guiding present in the laser structure, nr is independentof x. Therefore,

β = kr + jαm

2. (3.31)

Taking the square of (3.31) gives approximately

β2 ' k2r + jkrαm. (3.32)

The z-averaged effective wave number k(x) may be given the as a functionof ω and z-averaged carrier density N(x) [50]:

k(x) =ω

cn(ω,N(x)) + j

1

2[g(ω,N(x)) − αi] . (3.33)

Here n(ω,N(x)) is the modal index, g is the modal gain, and αi is the internalloss all of which are averaged over z. The internal loss includes losses causedby scattering of light at surfaces or at crystal defects, and by free carrierabsorption. For the modal gain we assume a simple linear model withoutany spectral dependence:

g ≡ g(ωr, N(x)) = Γa(N(x) −N0). (3.34)

Here a is the differential material gain and N0 is a reference carrier density.The relation between the modal gain and the material gain gm is g = Γgm.We expand the complex propagation constant around the reference frequencyωr and the transparency carrier density Nr = N0 + αi/(Γa) to first order:

k(x) = kr +∂k

∂ω(ω − ωr) +

∂k

∂N(N(x) −Nr). (3.35)

The gain in general also depends upon intensity through processes such asspectral hole burning and carrier heating. This effect of nonlinear gain maybe included by adding an expansion term in (3.35) proportional to the inten-sity. Normally, the expansion coefficient is negative since high intensity tendsto suppress the gain. Nonlinear gain is relevant at high powers. Modeling of

24

BA lasers using BPM has commonly implied use of linear gain models e.g.[30] [31]. Imitating a quantum-well gain with a nonlinear dependence on thecarrier density has also been used in conjunction with BPM [51]. A trendin the modern literature on time-domain methods for BA lasers is to relyon a microscopic treatment [3][4] of the gain material using the semiconduc-tor Maxwell-Bloch equations to describe spectral hole burning such that nophenomenological expression for the gain is needed nor is the introductionof the α-parameter described below. We shall see in Chapter 5 that theadiabatic elimination performed in connection with (3.5) has consequencesfor a system that includes diffraction of the field. By disregarding the fre-quency dependence of the gain, the direct-gap semiconductor is reduced toa two-level system without spectral broadening [45]. It is assumed that thecharge carriers of the semiconductor are in equilibrium, whereby excited elec-trons are mainly found at the bottom of the conduction band of the directbandgap semiconductor material. All events of generation and recombinationof electron-hole pairs are hence assumed for carriers of zero wavenumber. Inreality, the carriers are not in thermal equilibrium in a semiconductor laser.Furthermore, a linear dependence of the gain upon the carrier density is as-sumed, neglecting the effect of gain saturation at high pumping rates. Inthis thesis, we shall not consider cases for currents I > 1.2I0, where I0 isthe approximate threshold current. The reader may find a pump current20% above the threshold current rather modest. However, in a BA laser, atcurrents just above threshold, instabilities set in as we shall see in Chapters4 and 5. Here, the two expansion coefficients in (3.35) are taken to be

∂k

∂ω= 1/vg, (3.36)

where vg = c/ng is the group velocity, and

∂k

∂N=

1

2Γa(j − α). (3.37)

Here α is Henry’s linewidth enhancement factor [52] giving the couplingbetween the real and imaginary parts of the carrier-induced refractive indexchanges. The linewidth enhancement factor in BA lasers has experimentallybeen seen to vary with carrier density and wavelength [53]. One could includethis by adding higher order terms in the expansion of the wavenumber in(3.35). This, of course, urges that higher order expansion coefficients areavailable from measurements.

25

We move on to obtain our desired field equation. The square of k is givenas

k2(x) ' k2r + 2kr

[∂k

∂ω(ω − ωr) +

∂k

∂N(N(x) −Nr)

], (3.38)

when neglecting the terms quadratic in ∂k/∂ω or ∂k/∂N . We can now writethe field equations for Eω(x) using Eqs. (3.32) and (3.38) in (3.21)

∂2

∂x2Eω + 2kr

[∂k

∂ω(ω − ωr) +

∂k

∂N(N(x) −Nr) − j

αm

2

]Eω = fω. (3.39)

The field equation in the time-domain becomes∂2

∂x2− j

2kr

vg

∂

∂t+ 2kr

[∂k

∂ω(ωs − ωr) +

∂k

∂N(N(x, t) −Nr) − j

αm

2

]E(x, t) = f(x, t).

(3.40)We may define κ(x, t)

κ(x, t) = 2kr

[∂k

∂ω(ωs − ωr) +

∂k

∂N(N(x, t) −Nr) − j

αm

2

](3.41)

such that (3.40) becomes[∂2

∂x2− j

2kr

vg

∂

∂t+ κ(x, t)

]E(x, t) = f(x, t). (3.42)

The noise function f(x, t) is obtained from fω(x) via a transformation similarto (3.24).

Next, we must address the z-averaged carrier density. The mean fieldcarrier equation stated in the time-domain reads

∂

∂tN(x, t) = J (x, t) +D

∂2

∂x2N(x, t) − N(x, t)

τR− vggm(x, t)S(x, t). (3.43)

Here J (x, t) is the pump rate and D is the ambipolar diffusion coefficient.We assume that the pump rate is the sum J (x, t) = Js(x) + δJ(x, t) ofa stationary pump rate Js(x) and a small modulation-term δJ(x, t). In thetransverse y-direction the double heterostructure limits the diffusion of chargecarriers wherefore it is negligible in this direction. In the carrier equation,recombination of electron-hole pairs via processes other than stimulated emis-sion has been described through

R(N) =N

τR, (3.44)

26

where the carrier lifetime τR is assumed constant. Several recombinationmechanisms have thus been lumped together in the rate 1/τR. They arespontaneous emission, nonradiative emission, and possibly transverse leakageof carriers out of the active layer [54]. A more precise expression for R(N)involves a linear, a quadratic, and a cubic term in N .

We do not to include temperature effects in the analysis. For high pumprates the lateral temperature profile can certainly perturb the wave-guidingproperties of wide-aperture lasers [55]. A rise in temperature augments thereal part of the refractive index, which in turn causes thermal lensing. Inmodeling this is normally included by introducing a thermal index coefficient,which serves as a constant of proportionality between the temperature dis-tribution and the thermally induced change in the real refractive index. Thetemperature distribution can be found by solving the heat equation [40] orsimply by assuming a known temperature distribution [56]. In a more rigor-ous setting, the temperature affects microscopic properties, e.g. changes thebandgap of the active semiconductor, which in turn affects the macroscopicoptical properties [57]. Actual devices are mounted with heat sinks, and inexperiments thermal effects are often avoided by a slow temporal modulationof the pump current allowing for periodic cooling of the chip [20]. We shallassume that temperature effects are negligible in our calculations.

3.2 A single nonlinear field equation

In time-averaged measurements the near-fields of BA lasers can often befound to consist of a pedestal with a more or less regular ripple superimposedon top, see Figure 2.3 or e.g. [58]. That is to say that the near-fields are notdeeply modulated in the way a truncated sinusoidal is. Perhaps motivatedby such measurements Mehuys et al. [14] assumed a field solution of the formE(x) = E0 exp(a(x) + jφ(x)) underneath the metal contact with a(x) 1and φ(x) being real functions. Analytical approximate calculations involvinglinearizations give near-fields in rather good quantitative agreement withsome experiments at high currents. However, their starting point may bequestionable because a solitary BA laser operating at high pump currents isnot in a steady state. Measured near-fields that are only moderately and notstrongly spatially modulated are most likely a result of time-averages overmulti-lateral mode operation or alternatively a highly nonlinear (possiblychaotic) lateral variation in time and space. On the other hand, one can not

27

rule out that stationary solutions leave traces in time-averaged measurementsand from a fundamental point of view it is of interest to know about thestationary solutions of a physical system.

In order to perform a comprehensive study of the lateral modes structureof a BA laser and in addition to investigate stability properties as describedin the subsequent chapter, it is very helpful to derive a single equation thatincludes both the field and the carrier density [1]. The derivation is doneunder the assumption that carrier diffusion is negligible. This assumptionis argued for in [14]: The approximation is good as long as the condition ofk2

latL2D 1 is fulfilled; klat is the lateral wavenumber and Ld =

√DτR is the

diffusion length. However, our main reasons to exclude diffusion are given inthe following.

Our two main motivations to work with a single equation are: Firstly, inthe present chapter we shall show a series of newly found stationary solutions.We have searched for them like searching for needles in a haystack, notablyneedles whose existence we a priori were not aware of. Therefore a conve-nient computational environment has been a great advantage. Secondly, inChapter 4 we study the small signal properties of some of the calculated sta-tionary solutions. The mathematical apparatus derived there becomes rathercomplicated even without diffusion so leaving it out (for now) has been prac-tical. By no means, however, do we rule out the significance of lateral carrierdiffusion. In Chapter 5 the carrier diffusion is reintroduced in time-domaincalculations and in Chapter 6 also in stationary calculations.

Here we look for staionary solutions (Es(x), Ns(x), ωs). Stationary so-lutions are found as solutions to Eqs. (3.42) and (3.43) for a steady pumpterm J = Js(x) and for f(x, t) = 0. Upon neglecting the carrier diffusion

Ns(x) −N0 =Js(x)τR −N0

1 + |Es(x)|2/Psat

(3.45)

where

Psat =~ωh

2ε0nrcΓaτRK. (3.46)

In obtaining (3.46), Eq. (3.26) was used. The field Es(x, t) is seen to be inunits V/

√m. . Utilization of (3.45) in Eq. (3.41) leads to a single nonlinear

28

equation, namely∂2

∂x2+ 2kr

[∂k

∂ω(ωs − ωr) +

∂k

∂N

Js(x)τR −N0

1 + |Es(x)|2/Psat

− jαi + αm

2+ααi

2

]Es = 0.

(3.47)Note that the output power scales linearly with Psat. The second part of theoperator in (3.47) we define as

κs(x) = 2kr

[∂k

∂ω(ωs − ωr) +

∂k

∂N

Js(x)τR −N0

1 + |Es(x)|2/Psat

− jαi + αm

2+ααi

2

]

(3.48)and the field equation may simply be written

[∂2

∂x2+ κs(x)

]Es = 0. (3.49)

Let the field be defined on the interval −A ≤ x ≤ A. We must specify theboundary conditions. At a position on the x-axis that is sufficiently far awayfrom the metal contact for the intensity to become negligible, Eq. (3.49) canbe approximated to [

∂2

∂x2+ κWKB(x)

]Es = 0, (3.50)

with

κWKB = 2kr

[∂k

∂ω(ωs − ωr) +

∂k

∂N(Js(x)τR −Nr) − j

αm

2

]. (3.51)

For a slowly varying Js(x), one obtains the solution

Es(x) = E0 exp(±j√κWKBx). (3.52)

where E0 is a real constant. Assume that (3.50) is valid at ±A. Then atx = ±A the proper signs must be chosen to ensure a solution for the fieldthat decays exponentially when moving away from the metal contact. Thisgives us the derivatives at x = −A

∂Es

∂x= j

√κWKBEs(x) (3.53)

and∂Es

∂x= −j√κWKBEs(x) (3.54)

at x = A. We use Eqs. (3.52), (3.53), and (3.54) in the following to specifyboundary conditions.

29

J

Ax00−x0−A

Pump rate

Lateral position

Figure 3.1: Lateral distribution of pump rate. The points −A and A areboundary points

.

3.2.1 Current spreading

We must specify the profile of the pump rate Js(x). The pump rate is assumedto decay exponentially away from the current stripe, i.e.

Js(x) =

J exp((x+ x0)/d) forx < −x0

J for |x| < x0

J exp(−(x− x0)/d) forx > x0., (3.55)

where J is the pump rate underneath the metal contact and d is a currentdecay constant. The profile is illustrated in Figure 3.1.

As a unit for the pump rate we introduce J0. J0 is the approximatethreshold pump rate for the lowest order lateral mode:

J0 =1

τR

[Nr +

1

Γalln

(1

r1r2

)]. (3.56)

The relation between pump rate and current I is I = qV J where q is theelementary charge and V is the volume of the active region so that V = 2x0hl.

30

We use the terms pump rate and current interchangeably when distinctionis not important.

The idea of the form of Js(x) is that below threshold but above trans-parency, the lateral spontaneous emission profile can be measured. Pre-suming that the intensity distribution due to spontaneous emission is pro-portional to the carrier density, the decay of carrier density away from thepumped region may be fitted to an exponentional decay while using (3.45)with |E(x)|2 = 0 to obtain d [1]. When including the lateral carrier diffu-sion, it is common to let the diffusion spread the carrier density outside thepumped region. We do this in the next chapter. To obtain the carrier-densitydistribution in the active region in a more self-consistent way, procedures in-volving e.g. the Poisson equation can also be pursued [40][59]. The way onetreats the current spreading may certainly affect the obtained output field[60].

Solutions of the nonlinear field equation (3.49) with the profile of thepump rate (3.55) are now to be solved using numerical methods described inthe next section.

3.3 Numerical procedures for calculation of

modes

With the nonlinear differential equation along with boundary conditionsgiven below, we have a boundary value problem, which must be solvedthrough iterative methods.

3.3.1 Solutions with definite parity

Solutions for which |Es(−A)| = |Es(A)| and either Es(0) = 0 or dEs(0)/dx =0 possess definite parity. For such solutions we employ the numerical proce-dure of [1]. Due to the finite parity of the field distribution it is only necessaryto calculate the field on −A ≤ x ≤ 0. With an initial guess of a real vector(E0,L, ωs) the values of field and slope at x = −A become

Es(−A) = E0,L exp(−j√κWKBA), (3.57)

∂Es

∂x= j

√κWKBEs(−A). (3.58)

31

By (3.49), the field is propagated to x = 0 where it is evaluated. A Runge-Kutta method is used for this [61]. When searching for a symmetric solution,dEs(0)/dx = 0 is demanded while for an antisymmetric solution Es(0) = 0must be fulfilled. From the complex value of Es(0) or dEs(0)/dx, a Newton-Raphson routine suggests corrections to the values of (E0,L, ωs). With cor-rected (E0,L, ωs), the field is again propagated from x = −A. This iterativeprocess is repeated until (E0,L, ωs) has converged and a stationary solution(Es(x), ωs) is obtained by joining the appropriate part of Es(x) on 0 < x ≤ A.

3.3.2 Asymmetric solutions

Definite-parity solutions of nonlinear equations of the type 4u(x) + s(u(x))can, depending on the function s and the imposed boundary conditions some-times be shown to bifurcate into asymmetric solutions [62]. Asymmetric so-lutions are field distributions, which do not possess definite parity. In thiscase |Es(−A)| and |Es(A)| are in general not equal. Then, one must in thiscase calculate the field distribution on the entire domain −A ≤ x ≤ A. Wesplit the field into two parts. One part is propagated from x = −A with“initial conditions”

Es(−A) = E0,L exp(−j√κWKBA). (3.59)

∂Es

∂x= j

√κWKBEs(−A) (3.60)

to some point x1 which satisfies A < x1 < A. The other part is propagatedfrom the right (x = A) with

Es(A) = E0,R exp(−j√κWKBA). (3.61)

∂Es

∂x= −j√κWKBEs(A) (3.62)

also to x1. Let E0,R be a complex constant. At x = x1 we require the complexfield and its derivative to be continuous. This requirement renders a totalof four conditions. A four-dimensional Newton’s method gives corrections tothe four unknowns. A solution is found when the iteration converges to givethe four unknowns (ωs, E0,L,Re(E0,R), Im(E0,R)). The stationary solution isobtained by connecting the left and the right parts of the field which sharethe frequency ωs.

32

For definite-parity modes and in particular for asymmetric modes it iscrucial to have good initial guesses to obtain convergent solutions. Withinsufficiently good initial guesses either no solutions are found or the solverjumps to a solution far away form the wanted solution in case its existenceis known in advance.

3.3.3 Parameter values

Table 3.1: List of parameter values

Parameter Symbol Value Unit

Cavity length l 1.0 mm

Stripe width w 200 µm

Active layer thickness h 0.2 µm

Linewidth enhancement factor α 3.0

Linear gain coefficient a 1 × 10−20 m2

Confinement factor Γ 0.3

Effective refractive index nr 3.5

Effective group index ng 4.0

Reference wavelength λr 810 nm

Reference carrier density N0 1 × 1024 m−3

Internal loss αi 30 cm−1

Carrier lifetime τR 5 ns

Current decay distance d 10 µm

Left output power reflectivity r21 0.35

Right output power reflectivity r22 0.35

In this chapter we use the parameter values of Table 3.1. They effectivelyresemble those of [30][31] where a AlGaAs laser was investigated using BPM.We regard a stripe width of w = 2x0 = 200µm. We keep this width through-out the thesis. In the calculations presented in this chapter, the output facetswill be considered cleaved. As we are not regarding a specific device we havecalculatet neither the effective index nor the confinement factor but use the

33

values in the table. In actual high power devices most often the output facetis antireflection coated while the other facet is high-reflection coated. Inthis way practically all power is emitted through one of the mirrors only.Moreover, the intensity inside the cavity is reduced leading to less lateral fil-amentation and to a higher catastrophic power level [63]. For external cavityschemes, an antireflection-coating implies a higher level of feedback from theexternal reflector to the chip. This has been shown to improve the obtainedspatial coherence considerably. The angular dependence of reflectivity of acoated facet is enhanced as compared with a cleaved facet. This could beimplemented in a laser model, and is perhaps an overlooked issue in the(modern) literature on modeling of high-power lasers.

The justification of including only one longitudinal mode is based on argu-ments and measurements found in the literature, e.g. [1]. Spectrally resolvednear-fields show that the respective near-fields for individual longitudinalmodes appear similar. Similarly, the authors of [64] measured spectrally re-solved near-field intensity distributions of a freely running 100 micron wideBA laser. It can be seen that the individual longitudinal modes containsimilar lateral properties. That is to say that the read out from the grat-ing spectrometer which is a function of lateral position and wavelength issimilar for each longitudinal mode. It may hence be assumed that one canregard one longitudinal mode independently of the others. For a BA laserthat lack uniformity in the laser material, the assumption of similar lateralfield distributions for each longitudinal may become dubious [65].

3.4 Calculated stationary solutions

We find a systematic structure in the tuning curves, i.e. calculated solutionsin the current-frequency plane, and in the stationary solutions they represent.The systematic structure enables us to categorize different types of stationarysolutions. Figure 3.2 presents calculated tuning curves in the (J/J0, f)-planewhere f = (ωs − ωr)/(2π) is the relative frequency.

We categorize different types of field distributions Es(x) correspondingto different branches of the curves in Figure 3.2. The figure contains curvesfor 3 different types of modes. In addition to the 3 categories of modes inFigure 3.2, more exist as we shall see below. First, however, we regard thethree categories of Figure 3.2 named type I (mI), type II (mII), and typeIII (mII). m is an integer denoting the mode number. Our investigation

34

19

20

21

22

23

1 1.01 1.02 1.03 1.04 1.05

Rel

ativ

e fr

eque

ncy

[GH

z]

Pump rate J/J0

1I

2I

3I

4I

5I

6I

7I

8I

8II

7II

6II

5II

4II

3II

2II

4III

6III

8III

Figure 3.2: Tuning curves of modes-types mI , mII , and mIII for m = 1 tom = 8. Notice that the threshold pump rate of mI increases with m andthat mII and mIII emerge at increasingly high pump rate for increasing m.

focuses on rather modest pump rates. The reason for this is two-fold: Firstly,stability-analyses in the following chapter reveal instabilities at low currents.Secondly, we find a structure in the modes that roughly speaking tells thatwhat happens for a low-order mode also happens for a higher-order modebut at a higher current.

For all types of modes the integer m denotes the number of peaks in itsnear-field.

Type I modes

The first category of modes was also found in [1] albeit with different pa-rameter values. Type I modes are modes of definite parity. These modes arelabeled mI where m is an odd integer for symmetric modes and m is an eveninteger for antisymmetric modes. In Figure 3.2 these modes are seen to have

35

increasing threshold currents for increasing m caused by an increased absorp-tion in the unpumped layers as we shall review in Chapter 6. The lateraloutward flow of energy increases with increasing m causing larger losses atthreshold for higher order modes. At their respective thresholds mI are lineargain guided modes. When the pump rate is increased their field distributionsbecome perturbed due to the nonlinearity of the material and change shape,i.e. they do not merely increase in amplitude with an unchanged shape. InFigure 3.3 one can see near-fields for 6I at 4 different pump rates.

When following the almost horizontal parts of the mI-tuning curves fromthreshold and upwards in current, new branches emerge on the lower sides,e.g. at J = 1.018J0 for m = 8. These branches are tuning curves of thesecond kind of modes and will be described in the subsequent subsection.One characteristic feature of the type I modes is a dip in the middle of thenear-field which becomes deeper as the current is increased modestly abovethreshold. An example of this behavior can be seen in Figure 3.3 (a) and(b). This is studied in more detail in Chapter 6.

Further, the symmetric and antisymmetric modes differ in the sense thatthe symmetric modes (odd m) become increasingly compressed around x =0, whereas the intensity distribution of the antisymmetric modes (even m)becomes increasingly localized near the edges of the metal contact, i.e. nearx = ±x0 as the current is increased. Thus the intensity divides itself in twowhen the pump rate becomes considerable for a given mode mI for even m.Near-fields of 6I (antisymmetric) in Figure 3.3 and 5I (symmetric) in Figure3.4 for increasing pump rates show the general behavior of the 2 differentparities for m > 1.

The far-fields of 6I in Fig. 3.5 are representive of antisymmetric modes.Just above threshold (a), the far-field is nicely twin-lobed but as the currentis increased, nonlinear perturbations deteriorate the twin-lobes by addingmore and more structure around the lobes.

The change in far-field for the symmetric modes with increasing currentis partly different from the antisymmetric modes. For example, the far-fieldsof mode 5I shown in Fig. 3.6 reveal that just above threshold (a) and slightlyhigher above (b) the twin-lobe structure and the slighty perturbed twin-lobestructure, respectively, are similar to the antisymmetric modes. However,when increasing the current sufficiently the far-field becomes single-lobed asseen in (d). The single lobe is, however, perturbed by side lobes

It is sometimes said that BA lasers have twin-lobed far-fields. This is true

36

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10

20

30

40

50

-100 -50 0 50 100

Nea

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(d)

x [µm]

Figure 3.3: Near-field of mode 6I at different pump rates. (a) Just abovethreshold J = 1.0029J0; (b) Slightly higher pump rate J = 1.023J0. Notethe increased dip around x = 0; (c) J = 1.040J0; (d) J = 1.0675J0. Forhigher currents the intensity becomes more localized near x = ±x0.

for gain guided modes at threshold. However, when the current is increasedthis assumption breaks down since the modes become highly nonlinear. Thuseven if one could make the laser operate in a single lateral mode of type Ithe spatial coherence would be poor even for rather low currents. In Chapter4 we shall se that for m ≥ 3 all mI become unstable immediately abovethreshold or one can say that they are born unstable. Thus if a far-field ismeasured with a a twin-lobed structure it is most likely a result of the laserbeing in a time-dependent state over which a detector has averaged in time.Typically, at least for wide BA lasers with w ≥ 100µm, the time-averagedfar-field is a blurred shape and not a clear twin-lobe like in Figure 3.5 (a) or3.6 (a).

Type II modes (asymmetric)

The physical system under investigation is symmetric around x = 0. Yetwe find that stationary solutions without definite parity (asymmetric fielddistributions) exist due to the nonlinearity in the field equation and possi-bly also due to the boundary conditions. One type of asymmetric modesis presented in this subsection. This second kind of stationary solutions isdenoted mII . At the branch point where they are born in Figure 3.2, e.g.

37

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(c)

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0

10

20

30

40

50

60

-100 -50 0 50 100

Nea

r-fie

ld [A

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(d)

x [µm]

Figure 3.4: Near-field of mode 5I at different pump rates. (a) Just abovethreshold J = 1.0022J0; (b) Slightly higher pump rate J = 1.0199J0 Notethe increased dip around x = 0; (c) J = 1.0274J0; (d) J = 1.0665J0. For thehigher currents the intensity becomes compressed around x = 0.

J = 1.018J0 for m = 8, they have field distributions identical to the respec-tive definite-parity solutions mI , but for increasing currents they becomeincreasingly compressed on either side of x = 0. There are two f -degeneratesolutions for a point on the type II tuning curves: a left one and a right one.It can be seen that the tuning curve of mode mII eventually merges withthat of mode (2m)I when increasing the current. For example 4II mergeswith 8I at J ' 1.054J0, (a third branch is seen to branch out from this pointalso as explained in the next subsection). At the point where the mII-modemerges with the (2m)I-mode, the intensity distribution of the mII-mode islocalized on the left (right) side of the middle, i.e. on either −x0 < x < 0

or 0 < x < x0. At this point in the (J/J0, f)-plane the near-fields of thetwo degenerate mII-modes added together spatially overlap the near-field of(2m)I .

The evolution of a type II mode from its birth to higher currents is il-lustrated in Figures 3.7, 3.8, and 3.9. The near-field of mode 4II in Figure3.7 starts out identical to that of 4I around J = 1.0093J0. When increasingthe current, i.e. running along the tuning curve for 4II in Figure 3.2, thenear-field initially begins to tilt to either side (remember that asymmetricsolutions are doubly degenerate) as seen in 3.7. As the current is increasedfurther the field becomes more and more localized on either side of x = 0.

38

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ld [A

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(a)

Angle [°]

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Far

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(b)

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Far

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ld [A

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0.05

0.06

0.07

0.08

-3 -2 -1 0 1 2 3

Far

-fie

ld [A

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(d)

Angle [°]

Figure 3.5: Far-fields of mode 6I at the same pump rates as in Fig. 3.3. Asthe pump rates is increased, the twin-lobes become increasingly perturbed.

At the highest current represented in Figure 3.7, the near-field has left eitherside empty. For this current (J = 1.054J0) the tuning curve of 4II mergeswith the tuning curve of 8I . For even higher currents 4II becomes more lo-calized near x = ±x0. The far-field over the same current range as in Figure3.7 is presented in Figures 3.8 and 3.9 where it can be seen how the initiallysymmetric twin-lobed far-field becomes increasingly asymmetric and singlelobed. Note that the tilt in the far-field is opposite the one in the near-field.I.e. the angle of the dominant single-lobe in the far-field has the oppositesign of the overall slope of the near-field.

That asymmetric solutions of the field equation exist is not obvious, sincethe physical system is symmetric around x = 0. The fact that they emergefrom the branch of a mI-mode is due to saddle-node bifurcations as it willbecome clearer in Chapter 4. Perhaps mII-modes may be relevant in under-standing the way AEC lasers with their spatial filtering can operate becauseof the single-lobed far-fields of mII . However, also mII will in Chapter 4 befound unstable in such a way that they have to operate in a time-dependentstate. Note that 1II does exist but its tuning curve is not visible in Figure3.2 as it lies very close to the surrounding curves.

39

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ld [A

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ld [A

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(c)

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0.05

0.1

0.15

0.2

-3 -2 -1 0 1 2 3

Far

-fie

ld [A

.U]

(d)

Angle [°]

Figure 3.6: Far-fields of mode 5I at the same pump rates as in Fig. 3.4. Asthe pump rate is increased the twin-lobe change into a perturbed single-lobefield.

Type III modes (asymmetric)

In addition to modes of type I and type II, Figure 3.2 contains tuning curveslabeled mIII . These solutions also correspond to asymmetric field distribu-tions. Type III curves for odd m are not visible in the figure as they lie veryclose to their type I “parents” from which they emerge. At the point where anmIII-mode branches out from the mI -curve, the field distributions of mI andmIII are identical. Similarly to the behavior of type II modes, the near-fieldsof mIII become more and more compressed towards either side of x = 0 asthe pump rate is increased and they also have an f -degenerate solution whoseintensity distribution is the mirror image with respect to x = 0. Unlike mII

the near-fields of mIII do not tilt considerably as the current is increased. Toour findings only one asymmetric mode emerges from 1I, namely the modewe call 1II . Thus, apparently no mIII-mode exists for m = 1.

In Figure 3.10 it is evident how the near-field of 5III , representing odd m,is compressed towards either side of the pumped region when the current isaugmented. The corresponding far-fields in Figure 3.11 are single-lobed andbecome asymmetric only to a very little degree and are thus odd-m-mI-like.

The behavior of mIII for even m is slightly more complicated. The near-

40

-200 -150 -100 -50 0 50 100 150 200

x [µm]

1.015

1.025

1.035

1.045

Pump rate J/J0

0

5

10

15

20

25

30

35

40

A.U.

Figure 3.7: Mode 4II : Near-field for pump rates J = 1.0093J0 where themode emerges through J = 1.054J0 where the tuning curve of the modemerges with the one of 8I.

and far-fields of 4III are displayed in Figures 3.12 and 3.13. The fields fordifferent currents represent running along the 4III-tuning curve from thebranch point on 4I (a); on to a point where the tuning curve for 4III bends(b); on to a point after the bend (c); and finishing at a point far after thebend (d). It can be seen that the near-field remains centered around x = 0until finally sliding towards either side in (d). The far-field in Figure 3.13turns single-lobed although the parent 4I is never single-lobed in the far-field.Therefore, also mIII with m even seems odd-m-mI-like.

When the pump rate becomes relatively high and the intensity distribu-tion of mIII (for both even and odd m) is localized on either −x0 < x < 0or 0 < x < x0, its tuning curve merges with the tuning curve of yet anothertype of modes (labeled mIV ) that we have found to exist. This may be seenfor 4III in Figure 3.14, where tuning curves are shown.

41

-1.5-1-0.5

0 0.5

1 1.5

Angle [°]

1.0093 1.0097 1.0101 1.0105 1.0109

Pump rate J/J0

0

0.05

0.1

0.15

0.2

0.25

0.3

A.U.

Figure 3.8: Mode 4II : For pump rates J = 1.0093J0 where the mode emergesthrough J = 1.011J0. The far-field becomes increasingly single-lobed.

-2 -1 0 1 2Angle [°]

1.01

1.02

1.03

1.04

1.05

J/J0

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

A.U.

Figure 3.9: Mode 4II: For pump rates J = 1.011J0 through J = 1.054J0

where the tuning curve of the mode merges with the one of 8I . The dominantlobe moves outwards for increasing current.

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Figure 3.10: Mode 5III : Near-fields. (a) J = 1.0469J0 where mode emerges; (b) J = 1.0578J0; (c) J = 1.0684J0; (d) J = 1.0794J0. As the currentis increased the near-field is compressed towards either edge of the pumpedregion.

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Figure 3.11: Mode 5III : Far-fields. Same pump rates as in Figure 3.10.The far-field is centered at θ = 0 and becomes increasingly perturbed withincreasing pump rate. It is only slightly asymmetric even with a stronglyasymmetric near-field.

Type IV modes (definite parity)

We move on to present yet another exotic category of modes that we havefound to exist. So far we have seen one category of definite parity, namely

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(d)

x [µm]

Figure 3.12: Mode 4III : Near-fields.(a) J = 1.029J0; (b) J = 1.022J0; (c)J = 1.048J0; (d) J = 1.080J0. These four pump-rates correspond to followingthe tuning curve of 4III from where it emerges on 4I and around its bend.

0

0.05

0.1

0.15

0.2

0.25

-3 -2 -1 0 1 2 3

Far

-fie

ld [A

.U]

(a)

Angle [°]

0

0.05

0.1

0.15

0.2

0.25

-3 -2 -1 0 1 2 3

Far

-fie

ld [A

.U]

(b)

Angle [°]

0

0.1

0.2

0.3

0.4

0.5

0.6

-3 -2 -1 0 1 2 3

Far

-fie

ld [A

.U]

(c)

Angle [°]

0

0.1

0.2

0.3

0.4

0.5

0.6

-3 -2 -1 0 1 2 3

Far

-fie

ld [A

.U]

(d)

Angle [°]

Figure 3.13: Mode 4III : Far-fields. Same pump rates as in Figure 3.12.The far-field goes from being a perturbed twin-lobe to being a perturbedsingle-lobe. It is only slightly asymmetric even with a strongly asymmetricnear-field.

type I. For current levels close to their thresholds they can be viewed aslinear gain guided modes perturbed by nonlinear effects. We now come toa category of definite-parity modes which are solely nonlinear in the sameway that the asymmetric modes mII and mIII cannot exist without the

44

19

20

21

22

23

24

25

26

27

1.04 1.06 1.08 1.1 1.12 1.14 1.16

Rel

ativ

e fr

eque

ncy

[GH

z]

Current J/J0

4III

8IV

7IV

A

Figure 3.14: Tuning curves of modes 7IV , 8IV , and 4III . It can seen how 4III

merges with 8IV . The mark “A” corresponds to near- and far-field shown inFigure 3.15 (a) and (b).

nonlinearity of the gain material. The tuning curves in Fig. 3.14 each showinga loop and a cusp correspond to highly nonlinear finite-parity modes. Wedenote the modes corresponding to such tuning curves type IV. Examples for7IV and 8IV are seen in the figure. Again, the tuning curve for 4III is alsoseen merging with 8IV . When running along the tuning curves of mIV thefield distributions change considerably in character. On the upper parts ofthese curves (the high-frequency part) the near-field is a self-focusing solutionwhich is increasingly localized around x = 0 with increasing pump rate. Near-and far-field of 8IV near (J/J0 = 1.06, f = 22.3 GHz) (marked “A”) can beviewed in (a) and (b) of Figure 3.15. This is an example of a field distributionof the high-frequency part. It is rather remarkable that one can find self-focusing solutions in a system where the near-fields of the “conventional”modes (type I) have slowly varying dips around x = 0. So called spatialsolitons have been observed in BA optical amplifiers [66]. The spatial solitonswere traveling waves with self-focusing field distribution formed inside theBA amplifier. Possibly the self-focusing solutions, that we have found, arerelated to these experimentally observed solitons.

On the low frequency part of the tuning curve of 8IV the field distributions

45

0

20

40

60

80

100

-100 -50 0 50 100

Nea

r-fie

ld [A

.U]

(c)

x [µm]

0 0.2 0.4 0.6 0.8

1 1.2 1.4

-3 -2 -1 0 1 2 3

Far

-fie

ld [A

.U]

(d)

Angle [°]

0

20

40

60

80

100

-100 -50 0 50 100

Nea

r-fie

ld [A

.U]

(e)

x [µm]

0 0.1 0.2 0.3 0.4 0.5 0.6

-3 -2 -1 0 1 2 3

Far

-fie

ld [A

.U]

(f)

Angle [°]

0

10

20

30

40

50

-100 -50 0 50 100

Nea

r-fie

ld [A

.U]

(a)

x [µm]

0

0.1

0.2

0.3

0.4

0.5

-3 -2 -1 0 1 2 3

Far

-fie

ld [A

.U]

(b)

Angle [°]

Figure 3.15: Near- and far fields corresponding to points on tuning curves inFigure 3.14. (a) and (b) are near- and far-field corresponding to the mark “A”on tuning curve 8IV . (c) and (d) are near- and far-field of 8IV correspondingto the point where 4III merges with 8IV . (d) and (e) are near- and far-fieldof 4III at the same point.

46

are of a different character. In Figure 3.14 it is seen how 4III merges with8IV around J = 1.12J0. Figure 3.15 shows field distributions for 8IV ((c)and (d)) and 4III ((e) and (f)) at their merging point . At the point wheremIII and (2m)IV merge, the near-field of either mIII (there are two of them)makes up half the near-field of (2m)IV , whereas the far-fields are completelydifferent. To our findings, the lowest order mIV -mode is 3IV .

The systematic structure of the BA-laser modes

As can be clearly seen from Figure 3.2, the lateral modes in a BA laser havean underlying systematic structure. Type I modes are the basic stationarystates of BA lasers since they are linear gain guided modes at their respectivethresholds. The remaining types of modes can exist only due to the nonlin-earity of the gain material. The mode with the lowest threshold current is1I. Mode 1II branches out from 1I and merges with 2I . Mode 2II bifurcatesfrom 2I and merges with 4I . In addition mode 2III branches out from 2I ata higher current than 2II . Mode 2III also merges with a mode at a highercurrent, namely mode 4IV . Hence modes mII emerge from mI and eventuallymerge with (2m)I . Also, modes mIII emerge from mI and merge with modes(2m)IV . The exception is that “1III” does not exist.

19

20

21

22

23

24

25

1 1.01 1.02 1.03 1.04 1.05

Rel

ativ

e fr

eque

ncy

[GH

z]

Current J/J0

4I

8I

8II

4II

2II 4III

8III

•d,e

•f

•b,c

•a

Figure 3.16: Exert of 3.2 summarizing part of the systematic structure of theBA-modes. Modes 2II , 4I , 4II , 4III , 8I, 8II, and 8III are shown. The bulletscorrespond to near-fields in Figure 3.17 and far-fields in Figure 3.18.

47

0

10

20

30

40

50

-100 -50 0 50 100

Nea

r-fie

ld [A

.U]

(c)

x [µm]

0

10

20

30

40

50

-100 -50 0 50 100

Nea

r-fie

ld [A

.U]

(d)

x [µm]

0

10

20

30

40

50

-100 -50 0 50 100

Nea

r-fie

ld [A

.U]

(e)

x [µm]

0

10

20

30

40

50

-100 -50 0 50 100

Nea

r-fie

ld [A

.U]

(f)

x [µm]

0

10

20

30

40

50

-100 -50 0 50 100

Nea

r-fie

ld [A

.U]

(a)

x [µm]

0

10

20

30

40

50

-100 -50 0 50 100

Nea

r-fie

ld [A

.U]

(b)

x [µm]

Figure 3.17: Near-fields corresponding to bullets in Figure 3.16. (a) 4II ; (b)2II; (c) 4I; (d) 4II; (e) 8I ; (f) 8II.

48

0 0.02 0.04 0.06 0.08 0.1

0.12

-3 -2 -1 0 1 2 3

Far

-fie

ld [A

.U]

(c)

Angle [°]

0 0.05

0.1 0.15

0.2 0.25

0.3 0.35

0.4

-3 -2 -1 0 1 2 3

Far

-fie

ld [A

.U]

(d)

Angle [°]

0 0.02 0.04 0.06 0.08 0.1

0.12 0.14 0.16

-3 -2 -1 0 1 2 3

Far

-fie

ld [A

.U]

(e)

Angle [°]

0 0.2 0.4 0.6 0.8

1 1.2

-3 -2 -1 0 1 2 3

Far

-fie

ld [A

.U]

(f)

Angle [°]

0 0.1 0.2 0.3 0.4 0.5 0.6

-3 -2 -1 0 1 2 3

Far

-fie

ld [A

.U]

(a)

Angle [°]

0 0.02 0.04 0.06 0.08

0.1 0.12 0.14

-3 -2 -1 0 1 2 3

Far

-fie

ld [A

.U]

(b)

Angle [°]

Figure 3.18: Far-fields corresponding to bullets in Figure 3.16. (a) 4II; (b)2II; (c) 4I; (d) 4II; (e) 8I ; (f) 8II.

49

An exert of the structure with only tuning curves for 2II, 4I , 4II, 4III ,8I, 8II , and 8III is displayed in Figure 3.16. The points labeled (a)-(f) aredifferent examples of the just mentioned six modes at J = 1.054J0 (thecurrent at which 4II and 8I merge). The corresponding near-fields are foundin Figure 3.17 and the far-fields in Figure 3.18. In these two figures (a) is4III with a near-field that is asymmetric but almost untilted. Note that itsfar-field is single-lobed and centered around an angle of 0 degrees. (b) and(c) are 2II and 4I at a higher current than where 2II merges with 4I . Thenear-field of 2II is localized near −x0 or x0 and is seen to constitute “halfthe near-field” of 4I . Similarly, (d) and (e) are 4II and 8I at their mergingpoint. (f) is 8II.A close-up of 2III branching out from 2I is displayed in Figure 3.19. Alsohere 1II merges with 2I . This figure is then a close-up of Figure 3.2.

The remarkable systematics of the mode structure that we have found isdifferent from the mode structure of a DFB laser where symmetry-breakingfield solutions also exist. A multitude of solutions can indeed be found inthe case of a semiconductor DFB laser [67]. We believe that the structureapplies to all mode orders m as we checked for higher for orders, e.g. m = 15.Asymmetric modes may also be found in stripe geometry lasers [68].

Additional types of modes

In addition to the different modes mentioned so far, we have found two furthertypes of modes. They are asymmetric modes likely to bifurcate from mIV .We leave them to future work.

Light-current characteristics

The light current characteristics corresponding to the tuning curves in 3.2are displayed in Figure 3.20. Thus, displayed are light-current characteristicsfor mI , mII , and mIII for m equal to 1 through 8. The total output poweris given as (see Eq. (A.11))

Pout = 2ε0nrcαmlK

∫ ∞

−∞|Es(x)|2dx. (3.63)

As an example, modes 6I , 6II, and 6III are shown separately in Figure 3.21.As expected the branch points are located at the same pump rates as forcorresponding tuning curves.

50

19.3

19.32

19.34

19.36

19.38

19.4

19.42

19.44

1.015 1.0155 1.016 1.0165 1.017 1.0175 1.018 1.0185 1.019 1.0195

Rel

ativ

e fr

eque

ncy

[GH

z]

Pump rate J/J0

1I

2I

1II

2III

Figure 3.19: Tuning curve: Close-up of Figure 3.2. Mode 2III branches outfrom 2I and 1II merges with 2I .

3.5 Summary

In this chapter we have studied the lateral mode structure of a BA laser. Thestationary solutions were found using a mean field approximation, whichlead to equations in 1 spatial dimension. The simple lateral geometry ofa gain guided laser proved to contain a rich and systematic structure ofmodes and not only “standard” gain guided modes that we have labeled typeI modes (they in themselves show interesting properties). We introduced3 other categories of modes, and all four types of modes were seen to beinterrelated in spatial shape and through the structure of their tuning curves.Of particular interest were the asymmetric type II modes as their far-fieldsresemble the single-lobed far-field of the AEC laser presented in Chapter 2.In the following chapter we will test the stability of type I and type II.

51

0

10

20

30

40

50

60

1 1.01 1.02 1.03 1.04 1.05

Out

put p

ower

[mW

]

Pump rate J/J0

Figure 3.20: Light-current characteristics of modes mI , mII , and mIII for mgoing from 1 to 8. That is, Pout as a function of J/J0.

0

5

10

15

20

25

30

35

40

45

50

1 1.01 1.02 1.03 1.04 1.05 1.06

Out

put p

ower

[mW

]

Pump rate J/J0

6I

6II

6III

Figure 3.21: Light-current characteristics of modes 6I , 6II , and 6III .

52

Chapter 4

Small-signal analysis usingGreen’s functions: Stabilityproperties and responses

In the Chapter 3 we found stationary solutions using the nonlinear fieldequation (3.49). In this chapter we shall study the small-signal propertiesof some of them, namely type I and type II. First and foremost we want toperform a stability analysis. Lasers are nonlinear systems whose stationarysolutions in a small-signal analysis can be classified as stable or unstable.In semiconductor-laser theory it can sometimes be said that the stationarysolution having the lowest threshold gain of all modes is the dominant modeas the gain clamps at threshold. BA lasers are known to become pulsatingat very low continuous pump currents, showing multi-mode or even what islikely to be chaotic behavior and the gain of a given mode does not clamp atthreshold due to spatial hole burning. Thus, there is no indication that suchan assumption is generally valid for a BA laser.

In a small signal analysis, one may talk about two different kinds ofstability, namely local stability and global stability [67]. A locally stablemode is stable with respect to small-signal fluctuations. Fluctuations presentin the physical system will make a locally unstable mode go to a differentstationary solution or a time-dependent state. A local instability is alsocalled an instability of the saddle-point type. If a mode is locally stable it isnot necessarily globally stable. Other than the saddle-point-type instability,a mode may suffer from a self-pulsating instability, which must also be testedfor before a mode can be called globally stable. Specifically, if the system

53

determinantD(s) has two zeroes lying symmetrically around the real s-axis inthe right half s-plane, then the system is in a time-dependent state. This is aninstability of the Hopf-type. Here s = jΩ, where Ω is the baseband frequency.Instabilities of the saddle-point type, on the other hand, are associated witha zero of D(s) lying on the real s-axis in the right half s-plane. A saddle-point instability may be associated with a saddle-point bifurcation where azero of D(s) moves along the real s-axis from the left half s-plane into theright. Similarly an instability of the Hopf-type may appear in conjunctionwith a Hopf-bifurcation where the two complex conjugate zeroes of D(s)move from the left to the right half plane, e.g. when the pump current of thelaser is increased. The reason why we distinguish between, say, a pulsatinginstability of the Hopf-type and an actual Hopf bifurcation is that, as it shallbecome clear in this chapter, stationary solutions may be “born” unstable,such that an actual bifurcation cannot be detected; only the instability of amode is detectable in this case.

Laser models, where the photon and carrier density are effectively treatedwithout spatial dependence, usually yield an algebraic expression for the sys-tem determinant, whose zeroes in the complex frequency-plane are found bygraphical or iterative methods. The stationary solutions found in the previ-ous chapter have a spatial dependence in the lateral direction. To carry outa stability analysis on these modes, we employ a Green’s function method[69]. The method in its original form was derived for DFB lasers. For lat-eral gain guided modes, new small-signal equations must be derived. Forsaddle-point bifurcations we have, thanks to the work presented in the pre-vious chapter, the great advantage of knowing where new branches emergeon tuning curves. When the small-signal analysis “predicts” a saddle-pointbifurcation, perturbation theory tells us that one or more new branches ofsolutions will appear in a bifurcation diagram. We will be able to test theresults of the stability analysis by looking for one or more emerging brancheson the regarded tuning curve. When discussing bifurcations, intrinsicly onemust define one or more bifurcation parameters. When varying the bifurca-tion parameter(s) a given stationary state, or rather the dependent variablesassociated with it, can be followed along curves in e.g. the (J/J0, f)-plane,where J/J0 is the bifurcation parameter. In fact the pump rate will be theonly bifurcation parameter considered in the present thesis.

54

4.1 Small signal analysis using the logarith-

mic field

We start out by linearizing our field equation and carrier equation. We haveto perform the linearization of the two equations separately in order to intro-duce the spectral part of the carrier density properly. Our inhomogeneousfield equation in the time domain reads

[∂2

∂x2+ κ(x, t) − j

2kr

vg

∂

∂t

]E = f(x, t). (4.1)

Upon neglecting carrier diffusion the carrier distribution is given by

∂N(x, t)

∂t= J (x, t) − N(x, t)

τR− vggm(x, t)S(x, t), (4.2)

Finding the small-signal properties of various stationary solutions involves alinear expansion around a specific calculated mode. For convenience we firstintroduce the logarithmic field b:

b = lnE. (4.3)

It is noteworthy that Re(b) = ln|E| and Im(b) = φE, where φE is the phaseof E. The first and second derivatives of the logarithmic field are then

d

dxb =

1

E

d

dxE (4.4)

andd2

dx2b =

1

E

d2

dx2E − 1

E2

(d

dxE

)2

(4.5)

or from (4.4)

d2

dx2b =

1

E

d2

dx2E −

(d

dxb

)2

. (4.6)

Hence, from Eq. (4.1) the logarithmic wave equation is seen to be

d2

dx2b+

(d

dxb

)2

+ κ(x, t) − j2kr

vg

∂

∂tb =

f(x, t)

E. (4.7)

55

Next, we linearize the obtained logarithmic wave equation. Taking the dif-ferential on both sides of Eq. (4.7) yields

d2

dx2δb+ 2(

d

dxb)(

d

dxδb) + δκ− j

2kr

vg

∂

∂tδb =

f(x, t)

Es(x). (4.8)

Here Es(x) is a stationary solution. As κ is independent of Es(x), the differ-ential δκ is found to be

δκ(x, t) = 2kr

∂k

∂NδN(x, t). (4.9)

The small-signal expansion of (4.2) is found to be

∂

∂tδN = δJ − δN

τR− avgB

[δN |Es|2 + (Ns −N0)(E

∗sδE + EsδE

∗)]. (4.10)

From Eq. (4.3) we get δb = δE/Es and thus δE = δbEs and δE∗ = δb∗E∗s .

Substituting into Eq. (4.10) results in

∂

∂tδN = δJ − δN

τR− 1

PsatτR

[δN |Es|2 + 2(Ns −N0)|Es|2Re(δb)

]. (4.11)

Going to the baseband frequency domain implies taking the Fourier transform

sδN = δJ − δN

τR− 1

PsatτR

[δN |Es|2 + 2(Ns −N0)|Es|2Re(δb)

], (4.12)

since Es is stationary in time. A tilde over a symbol denotes that it is afunction in the baseband frequency Ω by s = jΩ. Solving for δN gives

δN(x, s) =δJ(x, s)τR

1 + sτR + |Es|2/Psat

− Js(x)τR −N0

1 + |Es|2/Psat

|Es|2/Psat

1 + sτR + |Es|2/Psat

2Re(δb).

(4.13)We divide the logarithmic field into real and imaginary parts by intro-

ducing the vector

ψ =

(Re(δb)Im(δb)

), (4.14)

implying Eq. (4.8) in a new form:

d2

dx2ψ +M1

d

dxψ + krΓaδN

(−α1

)+Mω

∂

∂tψ = f , (4.15)

56

where

M 1 = 2

(Re( db

dx) −Im( db

dx)

Im( dbdx

) Re( dbdx

)

), (4.16)

Mω =2kr

vg

(0 1−1 0

), (4.17)

and the noise is gathered in the vector f defined as

f =1

Es

(RefImf

). (4.18)

The Fourier transform of Eq. (4.15) yields the frequency domain equation

d2

dx2ψ +M 1

d

dxψ +Mψ = f + j (4.19)

where the matrix M is the sum

M = M 0 + sMω. (4.20)

ψ is the Fourier transform of ψ and

M 0 = 2krΓaJs(x)τR −N0

1 + |Es|2/Psat

|Es|2/Psat

1 + sτR + |Es|2/Psat

(α 0−1 0

). (4.21)

The current modulation term is given as

j = krΓaδJ(x, s)τR

1 + sτR + |Es|2/Psat

(α−1

). (4.22)

Now, Eq. (4.19) is the small-signal equation for the field and carrier density,

for which Eq. (4.13) has been utilized. Note that for Im(s) 6= 0, ψ is ingeneral complex.

Eq. (4.19) may be solved using Green’s functions [70]. For convenienceand for historical reasons we utilize the Green’s function of the adjoint of thedifferential operator of (4.19) instead. The operator

L =d2

dx2+M 1

d

dx+M (4.23)

57

has the adjoint operator

L† =d2

dx2− d

dxM

†1 +M †. (4.24)

Assume that ζi(x′, x, s) is the Green’s function for L†. Then

d2

dx′2ζi −

d

dx′(M †

1ζi) +M †ζi = eiδ(x′ − x) (4.25)

for i = 1, 2 and

e1 =

(10

)and e2 =

(01

). (4.26)

In the following we shall first derive an expression giving ψ for a givendriving term f + j. After deriving this general form of the response, weevaluate the Green’s functions ζi.

Partial integration of ζ†i(x′, x, s)Lψ(s, x′) gives

∫ A

−A

ζ†i (x

′, x, s)

(d2

dx′2+M 1

d

dx′+M

)ψ(x′, s)dx′ =

∫ A

−A

(d2

dx′2ζ†i (x

′, s) − d

dx′

(ζ†i(x

′, x, s)M 1

)+ ζ†i(x

′, x, s)M

)ψ(x′, s)dx′+

[ζ†i(x

′, x, s)d

dx′ψ(x′, s)

]A

−A

+[ζ†i (x

′, x, s)M 1ψ(x′, x, s)]A−A

−[d

dx′ζ†i(s, x

′, x)ψ(x′, s)

]A

−A

.

(4.27)

Alternatively, by manipulating the parenthesis in the right hand side inte-grand taking adjoints of both the interior and exterior, (4.27) may be put

∫ A

−A

ζ†i (x

′, x, s)

(d2

dx′2+M 1

d

dx′+M

)ψ(x′, s)dx′ =

∫ A

−A

(d2

dx′2ζi(x

′, x, s) − d

dx′

(M

†1ζi(x

′, x, s))

+M †ζi(x′, x, s)

)†

ψ(x′, s)dx′+

[ζ†i(x

′, x, s)d

dx′ψ(x′, s)

]A

−A

+[ζ†i (x

′, x, s)M 1ψ(s, x′)]A−A

−[d

dx′ζ†i (x

′, x, s)ψ(x′, s)

]A

−A

.

(4.28)

58

We demand that ζi fulfill the boundary conditions

d

dx′ζi(x

′, x, s) = M†1(x

′, s)ζi(x′, x, s) (4.29)

at x′ = ±A. In addition, it is presumed that the term dψ/dx is negligibleat x′ = ±A (see Appendix B). Then by inserting (4.19) and (4.25) in Eq.(4.28) we obtain

∫ A

−A

ζ†i (x

′, x, s)F dx′ =

∫ A

−A

δ(x′ − x)eTi ψ(x′, s)dx′ (4.30)

which leads to the general expression for the response

(ψ(x, s))i =

∫ A

−A

ζ†i (x

′, x, s)(f(x′, s) + j(x′, s))dx′. (4.31)

With Eq. (4.31) we have obtained the response for the noise f(x, s) and a

linear change of the pumping δJ(x, s). In other words the expression canbe used to calculate noise spectra and modulation responses. The notation“(ψ)i” means the ith component of the vector ψ.

The solution ζi of Eq. (4.25) can be found by solving the homogeneousequation on each of the two intervals −A < x′ < x and x < x′ < A. Byintegrating both sides of (4.25) on a small interval around x, the boundarycondition

[d

dx′ζi(x

′, x, s) −M †1(x

′, s)ζi(x′, x, s)

]x′=x+

x′=x−= ei, i = 1, 2 (4.32)

is obtained. We have used that M †ζi in (4.25) is continuous and made theinterval of integration arbitrarily small so that the integral over this termvanishes.

Inspired by the DFB-laser case in [69], we evaluate the Green’s functionnear the left boundary. Thus, we find the solution for x = −A + ε, whereε is an arbitrarily small distance. For simplicity we shall define ζi(x, s) =ζi(x,−A+, s) for x > −A, and denote ζ i(−A, s) the limit of ζi(x, s) forx→ −A+. Solving Eq. (4.25) with the conditions (4.29) for ζ i(−A, s) thencorresponds to solving the homogeneous equation

d2

dx2ζi −

d

dx(M †

1ζi) +M †ζi = 0 (4.33)

59

with the boundary conditions

d

dxζi −M †

1ζi = ei (4.34)

for x = −A andd

dxζi −M †

1ζi = 0 (4.35)

for x = A.Having formulated the homogeneous differential equation (4.33) with

boundary conditions (4.34) and (4.35) we set out to find relations for ζ i.Upon introducing the 4-dimensional vector

ui =

(ddxζi −M †

1ζi

ζi

)≡(vi

ζi

)(4.36)

and the 4 × 4 matrix

Mu =

(0 −M †

I M†1

), (4.37)

the homogeneous equation (4.33) may be put in the form

d

dxui = Muui. (4.38)

In Eq. (4.37) the entity 0 is the 2× 2 null matrix and I is the 2× 2 identitymatrix. We may also split Eq. (4.38) into the two equations

d

dxvi = −M †ζi (4.39)

andd

dxζi =M †

1ζi + vi. (4.40)

With the intention to solve Eq. (4.38) for ui we form the linear combination

ui =4∑

j=1

aijyj. (4.41)

One may also write (4.41) in matrix form

ui(x, s) = Y (x, s)ai(s), (4.42)

60

where Y is the matrixY = y1,y2,y3,y4 (4.43)

with column vectors yi. ai are column vectors with components

(ai)j = aij. (4.44)

From (4.38) and (4.42) the equation for Y is stated

d

dxY (x, s) = MuY (x, s). (4.45)

We choose the “initial conditions” at x = −A for Y to be

Y (−A, s) = I. (4.46)

Here I is the 4× 4 identity matrix. Equation (4.45) renders 4 linear coupleddifferential equations. With the condition (4.46) we can calculate Y (x, s)with a standard numerical routine for ordinary differential equations [61].The 4 vector functions yi obviously have linearly independent initial condi-tions (4.46) such that the linear combination in Eq. (4.41) is valid.

We introduce 4-dimensional unity vectors (ej)i = δij. From (4.34), (4.35),and (4.38) the boundary conditions at x = ±A for ui are seen to be

eT1ui = (ei)1 (4.47)

eT2ui = (ei)2 (4.48)

at x = −A andeT

1ui = 0 (4.49)

eT2ui = 0 (4.50)

at x = −A. In matrix form these boundary conditions become

Q(s)ai(s) = ei (4.51)

where

Q(s) =

eT1Y (−A, s)eT

2Y (−A, s)eT

1Y (A, s)eT

2Y (A, s)

. (4.52)

61

The shall need the adjoint of the matrix Q for the system determinant:

Q†(s) = q1, q2, q3, q4 (4.53)

with column vectorsq1(s) = Y †(−A, s)e1 (4.54)

q2(s) = Y †(−A, s)e2 (4.55)

q3(s) = Y †(A, s)e1 (4.56)

q4(s) = Y †(A, s)e2. (4.57)

Because of Eq. (4.46) Q becomes

Q(s) =

eT1

eT2

eT1Y (A, s)eT

2Y (A, s)

. (4.58)

The vectors ai(s) are calculated through

ai(s) = Q−1(s)ei i = 1, 2 (4.59)

whereby ui can be found. The system determinant is given as

D(s) = detQ†(s) (4.60)

which is simply

D(s) = (q3(s))3(q4(s))4 − (q3(s))4(q4(s))3. (4.61)

We have thus expressed the system determinant in terms of Y (A, s) whichcan be calculated numerically. By studying the complex zeroes of D(s) wewill be able to determine the stability properties of a given stationary solu-tion. The reason for the Hermitian conjugation “†” in (4.60) is that Q(s)appears in the denominator when calculating ai. Therefore it is the complexconjugate that appears in Eq. (4.31).

62

4.2 Local Stability

We introduce the stability parameter σ

σ =d

dsD(0). (4.62)

In Appendix C it is shown that for large positive s the system determinantD(s)is given as

D(s) ' 1

4s′e2

√2s′A (4.63)

where s′ = (2kr/vg)s. Then D(s) → +∞ for real s → +∞. Below, we alsoshow that D(s) has a fixed zero at s = 0. These two properties of D(s)imply the following: If σ < 0 then D(s) is negative in an interval (0, s0) onthe real s-axis and D(s) has an odd number of zeroes since D(s) → +∞for real s → +∞. Hence, if σ < 0 for a given stationary solution, it isunstable and its instability is of the saddle-point type. In case σ > 0, wedeclare the mode locally stable. If σ > 0, the given mode can, however, notbe declared globally stable since there may be zeroes of D(s) in the righthalf s-plane off the real axis. In principle, when σ > 0, D(s) may have aneven number of zeroes on the real axis in the right half s-plane. However,after doing some tests we have not seen this happening. Regardless, whenσ > 0 further investigations must be carried out before calling a mode ata given pump rate globally stable. A method to analyze global stability isdescribed in a subsequent section. When calculating σ as a function of thepump current a change in the sign of σ implies a saddle-point bifurcation.For σ = 0 we are at a bifurcation point where two or more tuning curvesmeet in a (J/J0, f)-diagram.

To make use of σ we must first show that D(s) has a fixed zero in s = 0.From Eq. (4.45) we obtain

d

dxY †(x, s) = Y †(x, s)M †

u (4.64)

The second column of M †u is seen to be

M †ue2 = −s2kr

vg

e3 (4.65)

Multiplying by e2 on both sides of (4.64) and then using (4.65) give

d

dxY †(x, s)e2 = −s2kr

vg

Y †(x, s)e3. (4.66)

63

In the limit of s = 0 it follows that

d

dxY †(x, s)e2 = 0 (4.67)

and Y †(x, s)e2 (i.e. the second row of Y ∗) is thus constant and equal to itsinitial value. That is,

q4(0) = Y †(−A, 0)e2 = e2 (4.68)

Of course when calculating Y numerically, this should hold true which weindeed find it to do. Inserting (4.68) in Eq. (4.61) leads to

D(0) = 0. (4.69)

Thus D(s) has a fixed zero at s = 0, and σ is a valid parameter decidingthe local stability of a calculated stationary solution. This makes it con-siderably easier to learn about the local stability of a mode, in fact we willnow derive a semianalytical expression for σ in terms of the matrix Q sothat a direct numerical calculation of the derivative dD(0)/ds using finitedifferences becomes unnecessary.

By Eqs. (4.61) and (4.68) we find

d

dsD(0) = (q3(0))3

d

ds(q4(0))4 − (q3(0))4

d

ds(q4(0))3 (4.70)

or by using (4.56) and (4.57)

d

dsD(0) = (y3(A, 0))1

d

ds(q4(0))4 − (y4(A, 0))1

d

ds(q4(0))3. (4.71)

To obtain the derivative ddsq4 for s = 0, we differentiate (4.66) with respect

to s and then take the limit s→ 0:

∂2

∂x∂sY †(x, 0)e2 =

−2kr

vg

Y †(x, 0)e3. (4.72)

Integrating Eq. (4.72) gives

∂

∂sY †(x, 0)e2

∣∣∣∣x=A

x=−A

= −2kr

vg

∫ A

−A

Y †(x, 0)e3dx. (4.73)

64

As Y †(x, s) = I at x = −A for all s, we use Eq. (4.57) to end up with

d

dsq4(0) = −2kr

vg

∫ A

−A

Y †(x, 0)e3dx = −2kr

vg

∫ A

−A

[(y1)3 (y2)3 (y3)3 (y4)3]†dx.

(4.74)The i’th component of the vector d

dsq4(0) is then

(d

dsq4(0)

)

i

= −2kr

vg

∫ A

−A

(yi(x, 0))∗3dx. (4.75)

With (4.75) the stability parameter σ can be calculated from Eq. (4.71)when y1 and y2 are known. Note that we have kept the complex conjugatein the integrand in Eq. (4.75) although yi are always real for s = 0. When asaddle-point bifurcation occurs σ is supposed to be zero [71]. Therefore whencalculating σ as a function of current we expect it to change sign when thetuning curve along which we are moving branches into one or more additionaltuning curves.

4.3 Calculation of y

We briefly comment on the calculation of yi. Antisymmetric stationary solu-tions have zeroes at x = 0. The matrixM 1 defined in Eq. (4.16) suffers froma singularity at x = 0 when Es(x) is antisymmetric as seen from Eq. (4.4).When calculating yi for these odd-parity solutions, we choose to smoothenthe “potential” given by M 1(x) at x. We do this by introducing the functionh(x)

h(x) =x

x + jεy

df

dx. (4.76)

Here the entity εy is a small real number. Obviously as εy → 0, h(x) → df/dx.Then for antisymmetric solutitions we replace df/dx by h(x) in M 1(x). Inh(x), evaluation of df/dx is still needed for h(0). We avoid this by averagingover the nearest discrete neighboring points: h(0) ' (h(∆xr) + h(−∆xl))/2.In this way a smooth potential is obtained. The small number εy is a freeparameter whose appropriate value is determined as follows: Denote thepump rate at which σ changes from positive to negative Jσ for a given εy.For an εy well large we find that σ will change from positive to negative at acurrent a bit too high as compared with the bifurcation point on the relevant

65

tuning curve. Decreasing εy will lead to a smaller value of Jσ. At some pointincreasing εy will not lead to a lower value of Jσ. We find that when thisoccurs, Jσ will equal the current at which the solution actually bifurcates,i.e. where a new branch emerges on the tuning curve. Typically εy ∼ 10−6

m.

4.4 Global stability

With a stationary solution whose σ > 0 it is necessary to evaluate its globalstability before naming it stable or unstable. In order to determine the globalstability of a mode at some current, we must know the signs of the real partsof the zeroes of the system determinant D(s). Instead of finding all zeroes ofD we define a real function of the imaginary part of s that will show a peakwhenever D(s) = 0. Regard the expansion around a zero of D(s) denoted s0

D(s) ' b(s− s0). (4.77)

Next, we introduce the ratio

1

D

∂D

∂s' 1

[s− jIm(s0)] − Re(s0)(4.78)

and state a function ρ(s), which is defined only for purely imaginary s, i.e.for Re(s) = 0:

ρ(s) = Re

1

D

∂D

∂s

. (4.79)

By comparison with (4.78)

ρ(s) ' −Re(s0)

[Im(s) − Im(s0)]2 + [Re(s0)]

2 . (4.80)

If ρ(s) is plotted, peaks in the curve will imply Im(s) = Im(s0). Thus atIm(s) = Im(s0)

ρ(s) ' − 1

Re(s0). (4.81)

Then for ρ(s = Im(s0)) < 0 for a given zero, the viewed mode is unstableat the given current as it implies a positive Re(s0). In contrast if ρ(s =Im(s0)) > 0 the zero does not contribute to an instability since Re(s0) < 0in this case.

66

The function ρ(s) is calculated numerically using Eq. (4.61) and a finitedifference is used to obtain the derivative ∂D

∂s. The derivative ∂D

∂sfor s → 0

calculated numerically should of course agree with σ. We find this to be thecase.

4.5 Results of stability analysis

4.5.1 Local stability of type I modes

In Figure 3.2 we saw how type II modes were born in conjunction withbranches emerging from the branches of type I modes. We expect a saddle-point bifurcation to occur when varying the bifurcation parameter (the pumprate) through a value where a branch point is located. However we cannot apriori tell which branches are locally stable or unstable. With the stabilityparameter σ, however, we are able to determine the local stability of a modeas a function of the bifurcation parameter. A type I mode undergoes asaddle-point bifurcation in conjunction with an emerging type II asymmetricmode. E.g. mode 5 bifurcates at approximately J = 1.0113J0. This familyof saddle-point bifurcations is seen to occur at an increasingly high currentfor increasing mode number. In Figure 3.2 there are 8 such bifurcations (form = 1 the 1II-branch is not visible as it lies close to the 1II-branch).

Figure 4.1 shows examples of calculated σ-curves for type I modes. Theregarded modes are 1I , 3I,4I , and 5I . The question we want the small-signalanalysis to answer is: on which side of the bifurcation point is σ > 0 yieldinglocal stability and on which side is σ < 0 yielding local instability. σ mustcross zero at the pump rate of the bifurcation point (where mII emerges) tobe in agreement with bifurcation theory.

Very reassuringly, indeed the σ-curves cross zero at the respective valuesof the pump rate that correspond to the values where the modes bifurcate andtype II modes emerge! Moreover, the actual result from the local stabilityanalysis is that σ > 0 for pump rates below the bifurcation point and σ < 0above the bifurcation point. We hence find that the mI-modes are locallystable on the interval of the pump rate going from threshold to the bifurcationpoint and lose their stability for higher pump rates. We have found this tobe the case for all mI with 1 ≤ m ≤ 8. In order to be able to conclude thatall mI -modes carry this behavior we also investigated mode 15I , for whichσ turns negative at 1.037J0. Again, this is the pump rate at which 15II

67

-4e+45

-3e+45

-2e+45

-1e+45

0

1e+45

2e+45

3e+45

4e+45

5e+45

6e+45

1 1.002 1.004 1.006 1.008 1.01 1.012 1.014Pump rate J/J0

σ×0.05, mode 3

σ×2.0, mode 4

σ, mode 5

σ×0.0003, mode 1

Figure 4.1: σ as a function of pump rate for modes 1I , 3I , 4I , and 5I . Thecurves cross zero where their corresponding stationary states saddle-pointbifurcate. This means that modes mI are locally stable for pump rates belowthe pump rate at which mII emerge.

emerges.We are rather confident that the above picture holds true for all m. Of

course no mathematical proof has been given here. Recall that mII is doublydegenerate for a given m; both a left- and a right-mII exist. Therefore two(not one) branches emerge from the bifurcation point. Consequently we canlabel the saddle-point bifurcations regarded here as pitchfork bifurcations[72].

4.5.2 Local stability of type II modes

Bifurcation theory predicts that the emerging type II branches correspondto locally stable modes mII since mI turned unstable for currents abovethe respective bifurcation points. A similar case where a symmetric solu-tion saddle-point bifurcates into an asymmetric one can be seen for DFBlasers [71]. At the bifurcation point where type I meets type II for some m,σ = 0 and their field distributions are identical. However, while σ becomesnegative for mI as the current is increased, σ for mII increases from zeroto an increasingly large positive value as the current becomes higher. Theasymmetric modes mII are hence locally stable. As an example, 6II , σ(J) ispresented in Fig. 4.2. The truncation of σ-curves at the lower-current end is

68

tricky as the fields of mI and mII become very similar near the bifurcationpoint. Regardless, we find that all mII are locally stable since σ(J) increasesuniformly. The result that type II modes are locally stable is interestingconsidering results of chapter 3. The far-fields of mII at currents well abovethe bifurcation points at which they emerge are single-lobed and relativelycoherent spatially. We note that we have not investigated the behavior of σnear the pump rates at which mII merge with (2m)II .

-5e+25

0

5e+25

1e+26

1.5e+26

2e+26

2.5e+26

1.013 1.014 1.015 1.016 1.017 1.018 1.019 1.02

σ

Pump rate J/J0

Figure 4.2: σ as function of pump rate. Above the pump rate of the bifur-cation point where the branch for 6II emerges from 6I , σ > 0 and uniformlyincreasing with increasing pump rate.

4.5.3 Global stability of type I modes

From the analysis of the local stability of modes mI we found above that thesemodes were stable for pump rates going from their respective thresholds tothe bifurcation point involving mII at which they underwent a saddle-pointbifurcation thus losing their local stability. In this current range the globalstability is now investigated using the function ρ(s) defined in Eq. (4.79).

Let us begin with the fundamental mode 1I for which the results can besummarized in short terms: Mode 1I is globally stable from its thresholdpump rate J = 1.0001J0 until it already Hopf-bifurcates at approximatelyJ = 1.0010J0. Note how 1I becomes unstable at a current that is higher thanthe threshold current of 2I . Also as a reference, 1I saddle-point bifurcatesat J ' 1.005J0. To illustrate how the Hopf-bifurcations occur, we show

69

ρ(s) for the case of 1I in Figure 4.3 whose abscissa is Im(s)/(2π). Graph 1corresponds to a pump rate of J = 1.00028J0. At this pump rate there are nopeaks below zero, and 1I is stable. Graph 2 represents a higher pump rateJ = 1.00153J0. Here two peaks have become negative corresponding to 2pairs of complex conjugate zeroes of D(s) moving into the right half s-plane.These two Hopf-bifurcations occur where Im(s)/(2π) is approximately 0.2GHz and 0.5 GHz. These two frequencies reveal side-mode coupling for 1I

to mode 2I and mode 3I, respectively, since the modespacings between 1I

and 2I and between 1I and 3I in Figure 3.2 lie very close to these Hopf-frequencies. Note that since the 0.2 GHz-peak is less sharp than the 0.5GHz-peak, the pair of zeroes corresponding to 0.2 GHz-peak have entered theright half s-plane for a lower current than the pair of zeroes corresponding tothe 0.5 GHz-peak. For a further augmentation of the current, more and morenegative peaks appears as seen from graph 3, which has 3 negative peaks.

Similar behavior is seen for mode 2I which is globally stable from itsthreshold at J = 1.00035J0 until it Hopf bifurcates at 1.00160J0. The evolu-tion of ρ(s) for 2I is given in Figure 4.3. Graph 1 represents J = 1.00053J0.Graph 2 is calculated at J = 1.00163J0 where a peak is seen to have turnednegative at 0.7 GHz. It should be noted that since 2I is antisymmetric wemust rely on the parameter εy. Therefore the current at which 2I bifurcatesand the frequency s of the instability may depend on the utilized value of εy.By comparing Figure 4.4 where the detuning of the zeros crossing the imag-inary s-axis is 0.7 GHz with Figure 3.2 where 4I lies 0.7 GHz above 2I tellsus that, most likely, the lowest-current Hopf-bifurcation of 2I is associatedwith side-mode coupling to 4I .

Modes m ≥ 3 lose their stability immediately above their thresholds.Alternatively they are all “born” unstable. As an example, the stability ofmode m = 3 was evaluated at ∼ 2 · 10−5J0 above its threshold where themode is clearly unstable with two negative peaks in ρ(s).

As an example of ρ for m ≥ 3, Figure 4.5 displays ρ(s) for mode 5I for 3different currents. Graph 1 is calculated at J = 1.00188J0 corresponding to7 · 10−5J0 above modal threshold. Graphs 2 and 3 represent higher currents,where more and more peaks become negative. For the sake of generality wehave in addition to the cases of 1 ≤ m ≤ 8 calculated ρ for the higher ordermode 15I and we find ρ to have several negative peaks immediately abovethreshold. Hence, we find good reason to believe that all modes of type mI

for m ≥ 3 are globally unstable immediately above threshold and most likelyremain unstable for all higher currents.

70

-2.5e-08

-2e-08

-1.5e-08

-1e-08

-5e-09

0

5e-09

1e-08

1.5e-08

2e-08

0 0.2 0.4 0.6 0.8 1 1.2 1.4

ρ

Frequency [GHz]

123

Figure 4.3: Mode 1I : ρ is plotted as a function of frequency Im(s)/(2π) for3 different pump rates. Graph 1: For J = 1.00028J0, ρ has no peaks withnegative values and the mode is therefore globally stable. Graph 2: ForJ = 1.00153J0, 2 peaks have become negative at 0.2 GHz and 0.5 GHz andthe mode is globally unstable. Graph 3: For J = 1.00247J0, a third peak hasbecome negative. When increasing the pump rate even further, more morepeaks will turn negative.

It is intuitively appealing that 1I does not lose its stability at a cur-rent where no other modes are above their threshold, since one associates aHopf-bifurcation with one or several emerging side-modes through side-modecoupling. There have been reports on experimental configurations designedto make BA lasers operate in the fundamental lateral mode only [64]. Theseschemes rely on making the losses of the fundamental mode considerablylower than those of the higher order modes by means of spatial filtering.With the notion that the fundamental mode cannot Hopf-bifurcate at a cur-rent below the threshold currents of the modes of higher order, such filteringschemes not only ensure that the fundamental mode be the only mode abovethreshold but also ensure its global stability at currents higher than the cur-rent at which it loses its stability in a solitary laser. It is also conceivablethat modes of higher order than 2 can be stable if by some means they arefiltered to have the lowest threshold current; see [73].

71

-3.5e-07

-3e-07

-2.5e-07

-2e-07

-1.5e-07

-1e-07

-5e-08

0

5e-08

0 0.2 0.4 0.6 0.8 1

ρ

Frequency [GHz]

12

Figure 4.4: Mode 2I : ρ is plotted as a function of frequency Im(s)/(2π)for 2 different pump rates. Graph 1: For J = 1.000531J0, ρ has no peakswith negative values and the mode is therefore globally stable. Graph 2:For J = 1.00163J0, a peak has turned negative at 0.7 GHz and the mode isglobally unstable.

We have seen that the fundamental mode 1I loses its global stability at avery low pump rate, namely 1.0010J0 (again, its threshold was 1.0001J0). In[25] a 100 micron-wide AlGaAs BA laser operating at 810 nm was investigatednear its threshold (recall that we study a laser of width w = 200µm). It wasfound that the single mode operation with the fundamental mode stoppedat a current 1.01I0 where instabilities set in. It must be noted that anexternal cavity with a plane mirror (no spatial filtering) was included in theset-up. However, a plane external mirror has not been found to stabilizelateral behavior in a BA laser [24]. Therefore we believe that it is relevantto compare our results with this experiment. Most likely the lateral mode-coupling becomes stronger with increasing width and a 200-micron wide lasermade from the same wafer as the device in [25] would suffer from an instabilityat a lower current when normalizing by the respective threshold currentsdue to the smaller lateral modespacing. We must also remember that theparameter values in our calculations are not taken from a specific device. Inany case, the experiment reports a very low pump rate as “threshold” for

72

-3e-08

-2.5e-08

-2e-08

-1.5e-08

-1e-08

-5e-09

0

5e-09

1e-08

0 0.5 1 1.5 2 2.5 3 3.5

ρ

Frequency [GHz]

123

Figure 4.5: Mode 5I : ρ is plotted as a function of frequency Im(s)/(2π) for3 different pump rates. Graph 1: For J = 1.001875J0, very close to modalthreshold, ρ has several negative peaks and the mode is therefore globallystable. Graph 2 and Graph 3: When increasing the pump rate more andmore peaks become negative.

lateral instabilities, suggesting that our small-signal analysis can give goodpredictions on the lateral modal behavior of a BA laser at very low pumprates.

4.5.4 Global stability of type II modes

We found earlier that modes mII were locally stable. However, we find alltype II modes to be globally unstable. For m > 1, ρ(s) behaves similarlyto ρ(s) for mI (for m ≥ 3). The mode 1II stands out to a small degree byhaving a low-frequency instability: at currents very close to the branch pointwhere 1II is born the mode only has a zero causing a low-frequency insta-bility at around 50 MHz. The explanation for this is most likely a couplingto 2II which lies 50 MHz above 1II in the tuning diagram. However alreadyat around J = 1.0065J0 a higher-frequency instability at 0.66 GHz appears;probably in conjunction with mode 4I according to the tuning curves. Re-gardless of this small abnormality, we have not found globally stable type II

73

modes.

4.5.5 Summary of stability properties

By investigating the small-signal stability properties of modes of type I andtype II we conclude that in a solitary BA laser with the parameter valuesused here, only modes 1I and 2I are globally stable at very low pump ratesonly. One experimental report for a 100 micron-wide BA laser confirms thatindeed stable, single-mode operation can only occur at very low pump rates.

All modes mI are locally stable for pump rates lower than the one atwhich a saddle-point bifurcation occurs in conjunction with the birth of mII .However they are globally unstable immediately above threshold for m ≥ 3.Modes mII were found to be globally unstable for all m. Since they arelocally stable they may in principle operate in some time-dependent state(just as mI may for pump rates where they are locally stable). Since mII

could be related to the time-averaged single-lobed far-field of the AEC laser,this is of our interest. However, since the AEC laser is a device that usuallyoperates at high pump rates, we can not conclude anything about this basedon our analysis at low currents.

Why do BA lasers in general not operate in a single lateral (per longi-tudinal) mode but rather in an either multimode state or a complex appar-ently chaotic state? In general terms the above stability analysis has giventhe small-signal answer to this question: All modes are unstable except forpumping at very low currents.

We have not investigated the stability of modes mIII and mIV . We leavethis to future work. Even if some of these modes are stable, which one couldintuitively doubt, then they will probably not be seen in a solitary BA laserwhere the large-signal behavior appears to be chaotic. The trajectory insome phase-space will never find a stable stationary solution when the BAlaser is well above threshold. In general one cannot expect that a laser willoperate in a cw-mode even if one or more stable stationary solutions exist.For example, in external cavity lasers (with one lateral mode) with strongoptical feedback, time-domain calculations show that limit-cycle operation,chaotic behavior, and mode hopping may occur in spite of the presence ofstable stationary solutions [74].

74

4.6 Linewidth

We move on to demonstrate the use of Eq. (4.31). In this section we willcalculate the linewidth for mode 1I. The linewidth is the low frequency limitof the frequency noise spectrum. To calculate small-signal noise spectra fromusing (4.31) one must know the diffusion matrix D(x, s) in the correlation

relation for f(x, s)⟨f(x, s)f

†(x, s)

⟩= D(x, s)δ(x− x′)2πδ(Ω − Ω′), (4.82)

where s = jΩ, s′ = jΩ′, and “〈〉” denotes ensemble averaging. In AppendixD the diffusion matrix is determined from the correlation relations for theLangevin noise function Fω(x, y, z) in Eq. (3.7). The diffusion matrix isapproximated to be independent of frequency

D ' Df(x, ωs)

2|Es|2I, (4.83)

where

Df(x, ωs) '2ω3

s~

c3ε0lKnrgnsp. (4.84)

The inversion factor nsp is given by [75]

nsp =N

N −N0. (4.85)

The Fourier transform of the instantaneous frequency deviation φE = d(ψ)2/dt

is s(ψ)2. The frequency noise spectrum is the spectral density of the instan-taneous frequency deviation

SφE(s) = lim

T→∞

1

T

⟨|s(ψ)2(s)|2

⟩(4.86)

By inserting (4.31) for i = 2 and j = 0 in Eq. (4.86) while using (4.82)one obtains the expression for the frequency spectrum at x = −A+

SφE(s) = |s|2

∫ ∞

−∞ζ†2(x, s)D(x, s)ζ2(x, s)dx. (4.87)

In the limit s→ 0 one obtains the spectral linewidth

∆ν = S(0)/2π. (4.88)

75

For the purpose of obtaining the linewidth we need to know lims→0 sζ2. Letus now derive an expression for ζ2 in terms of individual components of yi.From Eq. (4.42)

ζ2(x, s) =

(eT

3Y a2

eT4Y a2

). (4.89)

By Eq. (4.51) and (4.58)

a†2(s) = eT

2 (Q−1)† =1

D(s)

[0 1 (q3)

∗4(s) −(q3)

∗3(s)

]. (4.90)

and then

ζ2(x, s) =1

D(s)

((y2)3 + (y3)3 (y4(A))1 − (y4)3 (y3(A))1

(y2)4 + (y3)4 (y4(A))1 − (y4)4 (y3(A))1

). (4.91)

In (4.91) all yi are functions of s respectively x except those evaluated inx = A. The definition of σ in Eq. (4.62) gives

lims→0

sζ2(x, s) =1

σ

((y2)3 + (y3)3 (y4(A))1 − (y4)3 (y3(A))1

(y2)4 + (y3)4 (y4(A))1 − (y4)4 (y3(A))1

). (4.92)

In (4.92) all yi are functions of x except those evaluated in x = A. Havingobtained sζ2 in the limit s → 0 we can calculate the linewidth from (4.87)using (4.92) and (4.83). Figure 4.6 displays the calculated linewidth of 1I fortwo different pump rates. The linewidth at J = 1.009J0, just below the pumprate at which the mode Hopf-bifurcates is found to be ∆ν = 8.0 GHz. ForJ = 1.00028J0, close to its threshold J = 1.0001J0, the linewidth ∆ν = 33.0GHz. One expects the linewidth of a laser to drop for increasing current nearthreshold [76].

One could proceed to calculate the full frequency spectrum in (4.87) andother types of noise spectra. We have not succeeded in doing so as of yet.

4.7 Static Frequency tuning

With the general expression Eq. (4.31) one can also calculate the response

due to a current modulation δJ(x, s) by inserting the current modulation

term (4.22) in (4.31) and setting f = 0. The change in frequency when thecurrent is changed staticly is then given by

δω(0) = krΓa

∫ A

−A

lims→0

sζ†2(x, s)

(α−1

)τRδJ(x, s)

1 + |Es(x)|2/Psat

dx. (4.93)

76

5

10

15

20

25

30

35

1.0001 1.0002 1.0003 1.0004 1.0005 1.0006 1.0007 1.0008 1.0009 1.001

Line

wid

th [G

Hz]

Pump rate J/J0

Figure 4.6: The linewidth of mode 1I calculated for two pump rates. AtJ = 1.001J0, the mode Hopf-bifurcates.

where we again use the contracted notation ζ†2(x, s) ≡ ζ†2(x,−A+, s). UsingEq. (4.36)

ζ†2(s, x)

(α−1

)= α(u2(x, s))3 − (u2(x, s))4 = u†

2(x, s)(αe3 − e4) (4.94)

and then inserting the adjoint of u2 from Eq. (4.42) gives

ζ†2(x, s)

(α−1

)= a†

2Y†(x, s)(αe3 − e4). (4.95)

By Eq. (4.51) and (4.58)

a†2(s) = eT

2 (Q−1(s))† =1

D(s)

[0 1 (q3(s))

∗4 −(q3(s))

∗3

]. (4.96)

Using the definition of σ in Eq. (4.62) we obtain

lims→0

sa†2 =

1

σ

[0 1 (y4)1(A) −(y3)1(A)

]. (4.97)

Then it follows that,

lims→0

sζ†2(s, x)

(α−1

)=

σ−1 (y4)1(A) [α(y3)3(x) − (y3)4(x)] − (y3)1(A) [α(y4)3(x) − (y4)4(x)] .(4.98)

77

The small signal frequency tuning is then obtained by substituting (4.98)into (4.93).

-100

-50

0

50

100

1 1.01 1.02 1.03 1.04 1.05 1.06 1.07 1.08

∂ ω

/(2π)

/∂J’

[GH

z]

Normalized pump rate J/J0

Exact slope mode 4s.s.-slope mode 4

Exact slope mode 7s.s.-slope mode 7

Exact slope mode 15s.s.-slope mode 15

Figure 4.7: Comparison between small-signal frequency tuning denoted s.s.-slope and the exact slope of tuning curves denoted “Exact slope” for modes4I, 7I , and 15I in Fig. 3.2. J ′ is the normalized pump rate J/J0.

For a calculated stationary solution in the (J/J0, f)-plane it is thus possi-ble to calculate the local slope of the tuning curve by means of the small-signalanalysis. By local we mean at the current at which the stationary solutionwas calculated. We present a few examples of the calculated small-signalfrequency tuning as a function of pump current in Fig. 4.7. For modes 4,7, and 15 (all type I) we first directly calculated the slopes of actual tuningcurves in Fig. 3.2. These curves labeled “Exact slope” thus represent the“exact” slope of the tuning curves. The small-signal frequency tuning curvesas calculated using Eq. (4.93) are labeled “s.s.-slope”. We find that the pre-cision of the small-signal frequency tuning is very good for currents rangingfrom the threshold of modes mI to currents past the bifurcation point wheremodes mII are born. The discrepancy typically increases a little for largercurrents.

78

4.8 Summary

In this chapter we have performed a small-signal analysis on lateral modes ina BA laser. We employed a Green’s function approach to include the spatialdependence of the stationary solutions.

The main focus was on a stability analysis. The general result was brutalbut in correspondence with the known experimental and theoretical time-domain behavior of BA lasers: Except for the case of very low pump rates, allinvestigated modes were found unstable. Albeit brutal, the result is appealingfrom a theoretical point of view since the highly non-stationary output ofBA lasers conceivably has its origin in linear instabilities (i.e. instabilitiespredictable by a small-signal analysis). The only modes found to be globallystable were the two lowest order modes of type I. However, as the pump rateis increased from the threshold of the laser the fundamental mode 1I becomesunstable at already J = 1.0010J0 and the next higher order mode 2I losesits global instability at the slightly higher pump rate around J = 1.0016J0.All higher order type I modes are suffer from instabilities of the Hopf-typeimmediately above threshold. In connection with the birth of type II modes,type I modes bifurcates to also suffer from a saddle-point instability. Theasymmetric type II were found to suffer from instabilities of the Hopf-type.

We also demonstrated the calculation of the linewidth of the fundamentallateral mode and found plausible values for the linewidth of a single lateralmode just above threshold. Finally, we showed examples of frequency tuninggiving the change in oscillation frequency due to a static change in pumprate.

79

80

Chapter 5

Time-domain calculations

The output of BA lasers fluctuates in time and space even at small pumpcurrents. This is known experimentally [25] and from time-domain large-signal theory including microscopic descriptions of the gain material [3][4].Our analysis of the stationary states and their stability properties are incorrespondence with this known behavior; no stationary output from a BAlaser is possible except for at very low pump rates. When a stationary stateHopf-bifurcates into a limit cycle it could possibly be recognized in a time-domain simulation at a current just above the current of the bifurcation incase the limit cycle associated with the Hopf-bifurcation is stable. However,judging from the long time reported chaotic behavior of BA lasers, where noperfectly periodic mode of operation is seen for considerable currents, thisis only conceivable near threshold. In order to be able to compare resultsin chapters 3 and 4 with time-domain results, one could also do large-signaltime-domain calculations at very low currents and compare spectra obtainedfrom time-domain simulations (large-signal) with noise spectra (small-signal);in particular the field power spectrum. This was done successfully for a single-mode EC laser in [74]. We shall not pursue such a comparison here. Insteadwe turn to the time-domain considering a higher pump rate, where we seekto imitate the experimentally observed behavior of the asymmetric externalcavity (AEC) laser.

Of particular interest here is modeling of the AEC laser. In [73] BPMwas utilized for this. The field was propagated through the cavity of the chipand the external cavity until a steady-state was reached. However, at pumpcurrents slightly above threshold, the method becomes unstable, i.e. the fielddistribution differs from one round-trip to another. BPM being a method for

81

finding steady-state solutions, such a variation is not acceptable. However,the instability of BPM does add to the suspicion that even with the spatialfiltering in the external cavity the AEC laser is not a cw-laser. Thereforewe have pursued a time-domain approach. One could also try to extend themethods of chapters 3 and 4 to include an external cavity and perhaps eveninclude the spatial filtering, and then find stationary solutions. In addition astability analysis should be performed. However, recognizing the complexityof EC lasers in general this could become rather involved.

In the following we integrate the field- and carrier equations in the time-domain and for the AEC laser we add a filtered delay term in the fieldequation as it will be described. As mentioned in chapter 2, taking thephenomenological description of the semiconductor to the time-domain ina diffractive gain guided system is not without problems. Spatial Fouriercomponents of the field with large spatial frequencies may be amplified un-physically. We have experienced this first-hand by solving the coupled PDEsfor the field and the carrier density. The problem can be relieved by im-plementing an ad hoc part found in the literature in the field equation. Wediscuss this in more detail after the derivation of the needed equations. Theadvantage of the phenomenological approach as opposed to the microscopicapproach is a smaller computational load and also a more transparent de-scription of dynamic filamentation in BA lasers. Furthermore, to be able tocompare with our stationary results it is convenient to be able to use thesame parameters in the time domain. In the semiconductor Maxwell-Blochequations, one has to insert dephasing times as parameters. We note thatthe numerical scheme that we use to solve the PDEs for the field and car-rier density can be extended to include a microscopic treatment of the gainmaterial.

5.1 Time-domain equations

In the time-domain we now consider the one-dimensional field equation andcarrier-density equation that we obtained from the mean-field approximation.One may call these equations macroscopic. A set of similar one-dimensionalmacroscopic equations coupled to the semiconductor Maxwell-Bloch equa-tions applied to a BA laser were used in [22] with apparent success. Wetherefore believe that the one-dimensional treatment of BA lasers in the

82

time-domain generally is a good approximation.From Eq. (3.40) the time-domain field equation without external feedback

reads

j

2kr

∂2

∂x2E +

1

vg

∂

∂tE − 1 + jα

2Γa(N −Nr)E +

(αm

2

)E = f(x, t) (5.1)

where we have assumed ωs = ωr. We choose this frequency to obtain anequation similar to what is commonly found in the literature. In this chapterwe include carrier diffusion such that the carrier-density equation becomes

∂

∂tN(x, t) = J (x, t) +D

∂2

∂x2N − N(x, t)

τR− vggm(x, t)B|E(x, t)|2. (5.2)

In the present chapter the current-profile is a square, i.e. for |x| > x0 weset J = 0. Instead the carrier diffusion spreads the current. Further, thelateral boundary conditions are specified as

∂E

∂x= j

√κWKBE , x = −A (5.3)

and∂E

∂x= −j√κWKBE , x = A (5.4)

for the field. For the carrier density we utilize boundary conditions [77]

∂N

∂x= c2N , x = −A (5.5)

∂N

∂x= −c2N , x = A. (5.6)

The constant c2 is the ratio between the surface recombination velocity andthe diffusion coefficient, i.e. c2 = vsr/D [78]. Thus x = ±A are considered thelateral boundaries of the chip. We integrate Eqs. (5.1) and (5.2) numericallyusing the hopscotch method. We return to the subject of this method brieflyin Section 5.4 and in Appendix E. In the following two sections we derivean additional term describing external feedback that is to be included in thefield equation.

83

5.2 External feedback without filtering

We now continue to consider a BA laser with an external mirror placed likein Figure 2.2. To get started, we regard the case where the external mirrorprovides no filtering. The field equation (5.1) shall be extended to includethe feedback from an external reflector. Here we consider the case of smallto moderate levels of feedback.

When an external reflector is placed in front of the right facet, the effectiveright reflectivity is the reflectivity of a Fabry-Perot etalon [50]

rR(ω) =r2 + r3e

−jωτext

1 + r2r3e−jωτext(5.7)

Here r3 the effective external amplitude reflectivity. The effective reflectivityincludes reflection by the external mirror and any losses in the feedback path.τext is the round-trip time of the external cavity. If r3 is small Eq. (5.7) canbe approximated using the expansion 1/(1 + x) ' 1 − x:

rR(ω) = r2 + r3(1 − r22)e

−jωτext (5.8)

where terms with r3 of order higher than first have been neglected. Now,instead of the mirror loss of the solitary-laser field αm/2 = − ln r1r2/(2l) wemust consider the amplitude mirror loss of the compound cavity:

− 1

2lln r1rR = − 1

2l

[ln(r1r2) + ln(1 + γe−jωτext)

], (5.9)

where

γ =(1 − r2

2)r3r2

(5.10)

is the feedback parameter. In the limit of γ 1 the compound cavity mirrorloss can be approximated using ln(1 + x) ' x:

− 1

2lln r1rR ' − 1

2l(ln(r1r2) + γe−jωτext). (5.11)

Replacing the mirror loss of the solitary laser in Eq. (5.1) with the com-pound cavity loss in Eq. (5.11) yields the field equation with full (unfiltered)feedback

1

vg

∂

∂tE = − j

2kr

∂2

∂x2E +

1 + jα

2

∂g

∂N(N −Nr)E

− αm

2E +

1

2lγe−jωrτextE(−x, t− τext) + f(x, t).

(5.12)

84

Eq. (5.12), then, is the field equation introduced by Lang and Kobayahsi [79]except that diffraction is included in (5.12). Note, however, also the negativesign in front of x in the argument of E in the feedback term, as shall be ex-plained in section 5.3. It stems from the cavity of length 2f , which includesa lens in the middle and therefore two Fourier transforms. The diffraction inthe external cavity is thus included in (5.12).

From the view point of applied mathematics, differential equations in-cluding a temporally delayed term such as the term including E(t − τext)in (5.12) are called delay differential equations. Bifurcation theory and nu-merical bifurcation packages have been applied to single-mode EC lasers tostudy the birth of external cavity modes as well as their stability propertieswhen some bifurcation parameter is varied [80]. Of particular interest inthis thesis is the inclusion of the lateral dimension x. One possible path tofollow is to include the lateral field distribution as a fixed Gaussian shapewhose only degree of freedom is to change amplitude, that is to “breathe” asa function of time [81]. By doing so, however, one presumably dismisses thepossibility of any lateral instability, which we in Chapter 4 have found to beof great importance for BA lasers, and therefore the stability properties ofthe external cavity modes in Ref. [81] are very similar to those found withoutincluding the lateral dimension, i.e. similar to the stability properties of asingle-mode EC laser. Of course a reduced computational load is in favorof this approach, but it can only give insight in the behavior concerning thefundamental lateral mode.

Often the motivation for delayed feedback systems is to stabilize the fluc-tuating output of the regarded solitary system. If one a priori is aware of acharacteristic temporal period in an unstable nonlinear system, stabilizationof the system may possibly be achieved by using a delay of the known periodto obtain a stable motion [82].

5.3 Spatially filtered feedback

Unless some kind of spatial filtering is introduced in the feedback we can-not expect a stabilization of a BA laser. It has been seen experimentally in[24], where a BA laser was subject to an external mirror with no filtering,that no stabilization or spectral narrowing was achieved when compared toa solitary laser. In [83] a BA laser subject to a delayed spatially filtered

85

feedback was studied theoretically, where the filter function was a GaussianF (kx) = exp(−k2

x/ζ) for some constant ζ. However, the authors regarded aninfinite lateral structure, i.e. they imposed no lateral boundary conditions.Such a Gaussian filter has theoretically been shown to stabilize the field of theinfinitely wide semiconductor laser otherwise unstable without the externalfilter [84]. We saw in Chapter 3 that the stationary lateral mode structureis much affected by the lateral gain guiding wherefore the assumption of aninfinitely wide structure is mainly of theoretical or rather mathematical in-terest. As mentioned earlier, feedback from an external mirror with a finiteradius of curvature has been shown to have a stabilizing effect on BA lasersexperimentally [21] and theoretically [22] where the length of the delay wasof significance in addition to the spatial filtering.

We regard the filtering in the Fourier-plane of a laser, see Figure 2.2.A thin lens of focal length f is placed at (z = l + f) and an appropriatefiltering reflector is put at (z = l + 2f). The feedback term to be includedin the field equation should include propagation from chip to the lens; thephase added to the field by the lens; propagation from lens to the Fourier-plane; reflection and filtering due to the external mirror; and propagationback to the chip through the lens. The crucial thing here is to notice thatthe Fourier transform due to propagation from chip to the external mirrorhas the same direction (i.e. the same sign in the exponential function of theFourier integral) as the one due to the propagation from the external mirrorto the chip; the field sees the same lens twice. Thus the filtered field enteringthe chip becomes

Efb(x, t) = r3(1 − r22)e

−jωrτextFC(kx)F [E(x, t− τext)], (5.13)

where F denotes a forward Fourier transform and C(kx) is the filter functionin the kx-domain. Thus

Efb(x, t−τext) = r3(1−r22)e

−jωrτext

∫ ∫ ∫E(x′, t−τext)C(x′′)e−jkx(x′+x′′+x)dx′dx′′dkx

(5.14)leading to

Efb(x, t) = r3(1− r22)e

−jωrτext

∫ ∫E(x′, t− τext)C(x′′)δ(−x− x′ − x′′)dx′dx′′

(5.15)

86

and

Efb(x, t) = r3(1 − r22)e

−jωrτext

∫E(x′, t− τext)C(−x− x′)dx′ (5.16)

or

Efb(x, t) = r3(1 − r22)e

−jωrτext

∫E(−x− x′′, t− τext)C(x′′)dx′′. (5.17)

Note the minus in front of the unprimed x. This sign makes a very significantdifference. With a full plane mirror in the Fourier-plane C(kx) = 2π·constantone gets a delta-function filter function C(x). The minus sign in front of x inthe feedback term of (5.12) is justified, and it is in agreement with [30] whereBPM was used for a BA laser with unfiltered feedback. For an unfilteredfeedback, the minus sign in (5.12) says that the reflected field that reentersthe chip through the front facet, reenters at a position opposite to the pointat which it was emitted. Omitting the minus sign, i.e. convolving the filterfunction and field does not represent the optical system displayed in Figure2.2. In the case of a spatially filtered feedback we find that it causes verydifferent feedback conditions when the minus is left out. We shall return tothis point later in this chapter.

With a general spatial filtering we get the field equation

1

vg

∂

∂tE = − j

2kr

∂2

∂x2E +

1 + jα

2

∂g

∂N(N −Nr)E − αm

2E

+1

2lγe−jωrτext

∫ ∞

−∞C(−x− x′)E(x′, t− τext)dx

′ + f(x, t).

(5.18)

It should be noted that the filtering in Eq. (5.18) cannot be achieved withouta finite distance between the output facet and the filtering reflector becausethe two Fourier transforms are required. An optical system serves as anattractive environment for combining nonlinear phenomena with delay andfiltering.

5.3.1 Single stripe mirror

We consider the filter function relevant for the AEC laser. It represents asingle stripe mirror placed in the Fourier plane. Hence the width of the

87

2π

0Kx Kx+∆kxKx-∆kx0

Spatial frequency kx

Figure 5.1: Single stripe filter in the kx-domain.

mirror represents the range in spatial frequency or, alternatively, an angularrange. Regard the filter function representing a stripe mirror in the far-field

C(kx) = 2π1

2[sgn(Kx − kx + ∆kx) + sgn(−Kx + kx + ∆kx)] (5.19)

where sgn(x) is the signum function. Kx represents the center of the mirrorwhile ∆kx is the half width in spatial frequency. See Figure 5.1. This filteris the only type considered in this thesis. In position space we get

C(x) =ej(∆kx+Kx)x − e−j(∆kx−Kx)x

jx. (5.20)

Often one wants the angle in the Fourier plane:

θ = arcsinkx

k0. (5.21)

We acknowledge that the actual filter function may be more involved inan experiment. The width of the reflector is sometimes defined by two razorblades in front of the actual reflector. The adjustable distance between therazor blades then translates into a ∆kx. Such an arrangement may causeundesired scattering of light. However, we assume that such scattering lossesare included in r3.

88

5.4 Hopscotch method

The numerical method utilized in this chapter is the hopscotch method. Itis a method originally made to solve the parabolic equation ut = uxx [85].It is an explicit-implicit finite-difference scheme which has been found to bevery numerically stable [86]. The hopscotch method may also be used fornonlinear PDEs such as Korteweg-de Vries equation [87]. In our case, we areto solve the nonlinear system of PDEs in Eqs. (5.1) and (5.2). Details aregiven in Appendix E.

We find that the exact values of the “decay constants” in (5.3) through(5.6) are not important as it also found in e.g. [77]. If the boundary con-ditions on the x-domain are imposed well away from |x| = x0 the field andthe carrier density at the boundaries become very small. In an actual deviceone certainly wants the field to be vanishingly small at the lateral edges ofthe chip in order to avoid lateral lasing phenomena [88]. Also, it is conve-nient not having to know the actual magnitude of the surface recombinationrate. The noise term in the field equation drives the laser above threshold.We add Gaussian very-low level noise in the simulation. When the laser isabove threshold, the effect of the noise is negligible. For purely explicit in-tegration of the PDE Dut = uxx, one has the numerical stability criterion

D∆t/(∆x)2 < 1. For comparison we use vg/(2kr)∆t

(∆x)

2< 2.5 · 10−2. As al-

ready stated the macroscopic PDEs treated here can be used in conjunctionwith the semiconductor Maxwell-Bloch equations. Moreover, extending thehopscotch method to include z-dependence appears straight forward. Includ-ing the semiconductor Maxwell-Bloch equations and/or 2 spatial dimensionsare aimed for large-scale computing. For this purpose the hopscotch methodis very well suited for parallel computing [89].

5.5 The problem with adiabatic elimination

When integrating Eqs. (5.1) and (5.2) using the hopscotch method we haveexperienced first-hand that field components of high spatial frequencies maybecome amplified in an unphysical manner when the field is calculated witha relatively high spatial resolution. This behavior becomes clear when cal-culating the far-field, i.e. the spatial spectrum, in which the highest spatialfrequencies of the Fourier window grow orders of magnitude above the phys-

89

ical far-field localized within ±3 degrees around zero degrees (the laser axis).It turns out that the reason for this undesired behavior is not to be blamedon the hopscotch method; it is not a numerical instability. Rather the faultlies in the phenomenological description of the gain and amplitude-phasecoupling [41]. Due to the adiabatic elimination of the polarization variable,variations of high spatial and temporal frequency in the permittivity aredisregarded. A plane-wave linear stability analysis of equations similar toEqs. (5.1) and (5.2) shows that plane waves with high kx grow exponentiallywhen perturbed (they are unstable) [77]. Although this stability analysisdeals with a laterally infinite system, it captures that spatial Fourier com-ponents of the field with high spatial frequencies are inherently unstable.Again, this is not a physical result but a mathematical consequence of theintroduction of the simple linear gain model along with the α-parameter ina system with diffraction. As already implied, in a numerical integration of(5.1) and (5.2) with a relatively high resolution in x, i.e. a relatively small∆x, the Fourier window becomes large and Fourier components with a rela-tively high spatial frequency kx grow unphysically. As a result ∆x must bekept relatively large. A way to amend the high-kx instability while retainingthe phenomenological description has been suggested [41] [77]. The idea isto ad hoc introduce a small imaginary part in the factor 1/(2kr) in (5.1),i.e. 1/(2kr) → (1 + jε)/(2kr) with ε ≥ 0. The introduction of ε may beunderstood by transforming (5.1) into the kx-domain:

j + ε

2kr

k2xE +

1

vg

∂

∂tE − 1 + jα

2Γa(N −Nr)E +

(αm

2

)E = f(kx, t) (5.22)

It can seen that ε > 0 introduces a loss that depends parabolically on kx

whereby the components of high spatial frequency are dampened. Of coursethe effect on the physically relevant kx-values should be minimal when in-troducing ε. We have found that a value of ε = 7.5 · 10−3 does not raise thebackground carrier-density level significantly. With this value of ε we canrun calculations with ∆x = 1.33µm for a BA laser of width w = 200µmwhile keeping the high kx-components dampened for a pump rate J = 1.2J0.The spatial resolution reported in the streak-camera experiment in [2] was 3microns for a BA laser of a width of 100 microns (the time resolution was 50ps). Thus the obtained numerical resolution is acceptable. On the other handa higher resolution could be desirable for which a microscopic treatment ofthe semiconductor is necessary. In other words for a higher spatial resolutionone could couple the macroscopic PDEs to the semiconductor Maxwell-Bloch

90

equations through the polarization variable.At threshold the background carrier density is approximately 2.35N0 with

the utilized parameters. The additional loss due to the introduction of thesmall number ε must not significantly raise this level. One should be con-cerned with the background carrier density or alternatively the average car-rier density within |x| ≤ x0 or rather how much it differs from the carrierdensity outside the pumped region, as it is an important geometric factorhaving a significant influence on the shape of modes. This can be seen whenfinding stationary modes.

As an alternative to a microscopic treatment of the gain material onecould consider adding more terms in the expansion of the lateral wave numberin Eq. (3.35). If one included nonlinear gain in the phenomenological model,the problem of high-kx instability would, however, not necessarily be elimi-nated, as it seems to appear even at low powers. The problems associatedwith a phenomenological treatment of BA lasers in the time-domain reallyunderlines the complexity of the spatio-temporal behavior of BA lasers; wide-aperture multistripe index-guided laser arrays can be treated phenomenolog-ically without similar problems. As examples 1-dimensional modeling of alaser array of width 100 µm (10 stripes) [90] and similar 2-dimensional mod-eling of an array of width 50 µm (5 stripes) [86] were performed for highpump currents using linear gain and the α-parameter.

5.6 Time-domain results

Our motivation for doing time-domain simulations is to try and capture theexperimentally observed behavior of the AEC laser. First, however we regardthe freely running (solitary) laser as a reference. Then we turn to the AEClaser. We are mainly concerned with the post-relaxation behavior of thelaser since BA lasers are rarely used as switched devices. We change a fewparameter values as compared with Chapter 3. They are listed in Table 5.1.The length of the chip is now 500 µm, which is half of the length used inchapter 3. Further, we change the rear facet reflectivity to r1 = 1 so that wehave the same distributed mirror loss as in chapter 3. Thus the rear facetis now assumed anti-reflection coated, while the front facet is still assumedcleaved.

91

Table 5.1: List of parameter values

Parameter Symbol Value Unit

Cavity length l 500 µm

Stripe width w 200 µm

Active layer thickness h 0.2 µm

Linewidth enhancement factor α 3.0

Linear gain coefficient a 1 × 10−20 m2

Confinement factor Γ 0.3

Effective refractive index nr 3.5

Effective group index ng 4.0

Reference wavelenght λr 810 nm

Transparency carrier density N0 1 × 1024 m−3

Internal loss αi 30 cm−1

Carrier lifetime τR 5 ns

Front facet reflectivity r22 0.35

Rear facet reflectivity r21 1.0

Diffusion coefficent D 30 cm2s−1

External cavitity round-trip time τext 0.33 ns

Feedback parameter γ 0.15

5.6.1 Freely running laser

The dynamics of BA lasers has been studied extensively especially theoret-ically and perhaps to a lesser extend experimentally. Let us just let thephenomenological model demonstrate the characteristics of a solitary BAlaser. It captures the filamentation process qualitatively well. Even with amicroscopic treatment of the semiconductor gain material it is difficult to sayhow good the quantitative agreement with experiment is [2]. Figures 5.2 and5.3 show the turn-on behavior and the post-relaxation behavior of the totaloutput power and the laterally averaged carrier density as a function of time.The pump rate J = 1.2J0 is applied at t = 0. Following the relaxation oscil-lations, which are quickly “smeared out” due to lateral dynamics, the output

92

0

2000

4000

6000

8000

10000

0 10 20 30 40 50 60 70 80 90

Out

put p

ower

[mW

]

t [ns]

Figure 5.2: Output power as a function of time for freely running laser;J = 1.2J0. After the few relaxation peaks the laser enters a fluctuating stateand never finds a steady state.

power goes into a state where it fluctuates between 40 mW and 200 mW.The carrier density finds a background level NB(t) ' 2.4N0 around which itfluctuates but never finds a steady-state. One expects NB(t) to lie slightlyabove the steady-state threshold 2.35N0 due to spatial hole burning. Thisindicates that the introduction of the wavenumber-dependent loss throughthe small number ε, has not significantly changed the loss for the relevantkx-range.

The time-averaged near-fields reported from various measurements canbe described as a pedestal with a ripple on top. The time-averaged near-fieldin Figure 5.4 (a) shows those characteristics. For BA lasers of width, say,200 µm or more, the reported far-fields are usually blurred shapes localizedwithin an angular range depending on the width of the laser, the pumpcurrent, laser parameters, and design. Such a blurred shape is seen in Figure5.4 (b) where the time-averaged far-field corresponds to the near-field in (a).Lastly, the corresponding time-averaged carrier density is seen in (c). Notethe “ears” near the edges of the contact at ±x0. These are a consequense ofthe global guiding mechanism mentioned in Chapter 2; the real part of the

93

0

0.5

1

1.5

2

2.5

0 10 20 30 40 50 60 70 80 90

Bac

kgro

und

carr

ier

dens

ity N

B/N

0

t [ns]

Figure 5.3: Background carrier density as a function of time for freely runninglaser. Same case as Figure 5.2.

refractive index is relatively high in the middle of the active region.

94

0

20

40

60

80

100

120

-200 -150 -100 -50 0 50 100 150 200

Nea

r-fie

ld [A

.U]

x [µm]

0

2

4

6

8

10

12

14

16

-4 -2 0 2 4

Far

-fie

ld [A

.U]

Angle [°]

0

0.5

1

1.5

2

2.5

3

-200 -150 -100 -50 0 50 100 150 200

Car

rier

dens

ity N

/N0

x [µm]

(a) (b) (c)

Figure 5.4: Lateral distributions of freely running laser for J = 1.2J0. (a)Time-averaged near-field; (b) time-averaged far-field; (c) time-averaged car-rier density. The averages were made over 60 ns.

Time-averaged fields only tell us a small part of how the laser operates.The complexity of the dynamics of a BA laser is demonstrated in Figures5.5 and 5.6 where the near-field and far-field, respectively, are shown overa duration of 10 ns. The evolution of the near-field shows how filamentsmove laterally as a function of time and change direction randomly caus-ing the characteristic zig-zag patterns. This behavior is also found with amicroscopic description of the semiconductor. The far-field shows a highlyirregular behavior very far from a steady pattern.

5.6.2 Asymmetric external cavity laser

The motivation for the AEC laser is an improved spatial coherence or alter-natively a higher beam-quality as compared with the solitary BA laser. Thesingle lobe of the output can, at least after additional spatial filtering hasbeen performed, resemble a Gaussian beam. Our aim here is to reproduce,numerically, what we believe is the basic mode of operation for the AEClaser.

Spatial coherence has been shown to depend on the position of the stripemirror in the far-field [34]. In other words there is a position of the stripemirror that gives optimum spatial coherence for a given pump current. Morespecifically this optimum position gives the lowest full width at half maximumof the single lope and least possible intensity at other angles in the far-field.Experimentally it is seen that increasing the pump current invokes a needof adjusting the stripe-mirror in an outward direction to retain optimumspatial coherence [91]. However, again, this observation is based on time-

95

0

100

200

300

400

500

600

700

800

900

-200-150-100 -50 0 50 100 150 200

x [µm]

50

52

54

56

58

60t [

ns]

Figure 5.5: Near-field emission over 10 ns of freely running laser for J = 1.2J0.Dynamic filamentantion causes zig-zag patterns.

96

0

20

40

60

80

100

120

-4 -3 -2 -1 0 1 2 3 4

Angle [°]

50

52

54

56

58

60t [

ns]

Figure 5.6: Far-field emission over 10 ns of freely running laser for same caseas Figure 5.5.

97

averaged measurements and therefore it is difficult to tell what an increasedcurrent actually causes. Notably, we saw in Chapter 3 that for the stationarysolutions mII , the lobes in the far-field moved outwards (towards higherangles) as the current was increased.

We are not aware of the actual feedback levels due to the external mirrorin AEC lasers; they can conceivably be obtained by measurements, however[24] [92]. For single-mode EC lasers, there exist different regimes of behaviorin terms of stability and linewidth depending on, among other things, thefeedback level; see e.g. [42]. No similar identification of regimes exists forAEC lasers.

Based on the perspective of this thesis, which is inclined towards spatialproperties more than spectral properties, feedback levels of external cavityspatial filters can be divided into (at least) two regimes. We have seen inChapter 3 that asymmetric field distributions exist in the regarded symmet-ric system. Since mII-modes have single-lobed far-fields, they are interestingwhen regarding AEC lasers. If the asymmetric filtered feedback is relativelyweak, one can expect that it “helps” the laser to operate in such an asymmet-ric state by introducing a small asymmetry and by lowering the mirror loss forthe particular mII . This regime is mainly likely at low pump currents. Thesecond regime would be a regime where the reflectivity of the external stripemirror is so large that the regarded physical system (the chip and externalcavity) possesses a considerable asymmetry so that also linear solutions tothe field equation could be highly asymmetric. This second regime would berelevant for high currents.

Measurements of output-intensity distributions of AEC lasers with mirrorreflectors have to our knowledge all been time-averaged. Considering thedynamics of solitary BA lasers it is somewhat questionable that placing astripe in the Fourier plane should completely stabilize the laser temporally.Spectral measurements have shown that the spectrum of an AEC laser witha mirror reflector has roughly the same width as the spectrum for the solitarylaser [37]. With a grating reflector the reduction of the width of the spectrumis about an order of magnitude (in wavelength) [35]. The role of the finitelength of the external cavity is not clear as no experiments have been reportedwhere a systematic investigation of the dependence of the AEC laser on eitherthe length of the external cavity or the feedback level have been performed.

While the role of the length of the external cavity is unclear, it has beenestablished that in the Fourier-plane of the laser one angular position (or

98

rather two ±θ due to symmetry) is optimum for a given pump current.Again, optimum means an angular position of the external reflector at whicha single-lobe on the opposite side of the laser axis in the far field is enhancedas much as possible relative to the remaining parts the far-field. As it willbe shown below, we do not find that this optimum position of the mirrortranslates into a field in a stationary state at a pump rate of J = 1.2J0. Wecan, however, not rule out the existence of stable stationary solutions for anAEC laser. To be able to do so, one would have to include the full fieldequation with filtered delay in a stability analysis. However, this is probablymainly relevant at very low pump currents.

0

10

20

30

40

50

-4 -2 0 2 4

Nea

r-fie

ld [A

.U]

Angle [°]

Figure 5.7: Far-field (time averaged) of asymmetric external cavity laser forJ = 1.2J0. The stripe-mirror has been centered at angle θ = 0.832. Thesingle lobe peaks at θ = −0.96. The average was made over 60 ns.

Fig. 5.7 illustrates a time-averaged single-lobed far-field for a AEC laserwhere the angular position of the external mirror is close to optimum. Themain peak of the far-field is located at θ = −0.96. The laser is again pumpedat J = 1.2J0 and the time average was performed over 60 ns from t = 50 nsto t = 110 ns. The external mirror is centered at θ = 0.832. The angularspan of the mirror is 2∆θ = .59, which is similar to the values in the originalpaper on AEC lasers [29]. The feedback parameter γ has the value 0.15 whichis the value used in the original paper by Lang and Kobayashi [79]. Withr22 = 0.35 this value of γ corresponds to r2

3 = 1.86 · 10−2. The length of theexternal cavity is 5 cm equivalent to τext = 0.33 ns. Recall that measuring a

99

far-field like the calculated one in Figure 5.7 requires that a beam-splitter isinserted in the cavity just just before the stripe mirror in the Fourier planeto get the entire far-field, i.e. both negative and positive angles. We findthat the far-field in Figure 5.7 is in a good qualitative agreement with theexample of a measured far-field in Figure 2.4. We will discuss the agreementfurther after looking at the near-field and carrier density.

0

20

40

60

80

100

120

140

-200 -150 -100 -50 0 50 100 150 200

Nea

r-fie

ld [A

.U]

x [µm]

Figure 5.8: Near-field (time-averaged) of asymmetric external cavity laser.Same case as in Figure 5.7. Note how the field tilts to theright, while thesingle-lobe in the far-field is located at an negative angle.

The time-averaged near-field can be seen in Figure 5.8. It has 8-10 peaks.Note how the near-field tilts to the right, i.e. the left edge of the field ishigher than the right edge of the field. It thus tilts to the opposite side ofthe single lobe in the far-field. The orientation of the tilt is in agreementwith the experiment of [33] shown in Figures 2.3 and 2.4, where the sameorientation of the near-field compared to the far-field was seen.

Due to the intensity distribution in the time-averaged near-field, the time-averaged carrier-density distribution in Figure 5.9 forms a “cradle” in thecarrier density on the left side implying a nonlinear waveguide since therefractive index is relatively high where the carrier density is relatively low.We note that the tilt in the near-field is opposite to the one we found for themodes mII in Chapter 3. The near-field of mII with its single lobe at, say, anegative angle, has a positive “global” slope, see e.g. 8II in Figures 3.17 (f)and 3.18 (f). However, in the present time-domain calculation we are at aconsiderably higher current, and we have to remember that mII were found

100

0

0.5

1

1.5

2

2.5

3

-200 -150 -100 -50 0 50 100 150 200

Car

rier

dens

ity N

/N0

x [µm]

Figure 5.9: Carrier-density distribution (time-averaged) of asymmetric ex-ternal cavity laser. Same case as in Figure 5.8.

to be globally unstable.For an AEC laser it was shown experimentally in [73] that for pump

currents close to threshold it was possible to pick out lateral modes (due tothe external stripe-mirror) that resemble mII-modes, i.e. the single lobe inthe far-field and the edge of the near-field with highest intensity were orientedoppositely. When they increased the current, however, the situation changedto the one we find in this chapter and that was found in [33]: the single-lobein the far-field and the edge with highest intensity in the near-field have sameorientation.

In [34] the peak angle of the single lobe was 2.1 degrees for a 200 micronwide laser lasing at 810 nm and driven at 2 times threshold. In experiments,the angle at which the single lobe is centered moves outwards as the currentis increased [91]. Our calculation at 1.2 times threshold gives a smaller valuethan 2.1 degrees as expected. In any case we would not expect to be able totheoretically obtain exact agreement with an experiment without knowingrelevant parameter values of that very experiment.

In experiments it is seen that the output-beam quality of the AEC laserbenefits greatly from using an anti-reflection (AR) coated output facet. Powerreflectivities as low as 10−5 are obtainable [73]. We cannot readily includesuch low reflectivities in the Lang-Kobayashi-type field equation. When com-paring the time-averred far-field in Fig. 5.7 with an experimental one wenotice that the pedestal on the left side of the single lope is higher than

101

0

200

400

600

800

1000

1200

1400

-200-150-100 -50 0 50 100 150 200

x [µm]

75

80

85

90

95

100

t [ns

]

Figure 5.10: Near-field emission over 25 ns of asymmetric external cavitylaser for J = 1.2J0. The zig-zagging of filaments has been inhibited due tothe external mirror.

102

what is usually seen experimentally. We partly owe this to the lack of ARcoating in the simulation. Moreover, we could probably perform additionaloptimization in our calculations by adjusting the width ∆kx of the externalmirror. We have not optimized this parameter. Nevertheless, we are ratherconfident that the mode of operation, which we have found numerically, isthe one seen experimentally. Particularly because there exists only one op-timum angular position, which is rather tricky to find, where the AEC laserbehaves optimally single-lobed. We find this to be the case numerically as itis found experimentally [91].

We now turn to the dynamics underlying the time-averaged behavior.Figure 5.10 shows a 25 ns interval of the near-field. We see how the lat-eral movement of filaments has been almost stopped. Instead the filamentsnow more tend to stick to each their “lane”. This behavior resembles thedynamics of a lasr array. We also see that the nice coherent parts of thetime series are actually located on the right side despite the fact that in thetime-averaged near-field we saw that the intensity was globally higher onthe left side. However violent bursts of intensity to the left of the middlecauses the tilt in the time-averaged near-field. Therefore a relation to themII-modes is perhaps conceivable. However, the spatio-temporal evolutionis still irregular..

For applications the far-field is the most relevant and naturally the tem-poral irregularities of the near-field translate to the far-field in Figure 5.11.The main intensity peaks are around −1 as expected from the time-averagedfar-field. The time variation is still complex in spite of filtering provided bythe external stripe-mirror.

There is thus no indication that the stripe-mirror in the external cavitycan force the behavior on the chip into a steady state in spite of the improvedspatial coherence. Nor is there a clear sign of a characteristic time periodof the system to which τext can be tuned to match. When looking closelyat the intensity at x = 0 one can see fluctuatuins close to τext, but the timevariation is not periodic. Recently, (periodic) picosecond-pulse generation ina BA laser was demonstrated experimentally with an AEC-like laser whichhad a grating as the reflector in the external cavity [93]. The BA laser inthe experiment was AR-coated. It is presumable that a grating reflector isattractive to force the laser into a state that is periodic in time.

We briefly return to the discussion of the sign in the convolution in con-junction with a stripe-mirror filter. By omiting the minus sign, we find that it

103

0

50

100

150

200

250

300

350

400

-4 -3 -2 -1 0 1 2 3 4

Angle [°]

75

80

85

90

95

100

t [ns

]

Figure 5.11: Far-field emission over 25 ns of asymmetric external cavity laser.Same case as in Figure 5.10. Main intensity peaks appear around θ = −1

corresponding to the time-average.104

is easily possible to obtain perfect (time-averaged) single-lobe far-field. Thefar-field is thus localized tightly around a specific angle. The time variationcan be made perfectly periodic with τext as one of the characteristic periods.Also, in terms of phase-plane plots, one can obtain beautiful toruses like it ispossible in single-lateral-mode EC lasers [80]. In essence the AEC laser be-haves much more like a single-lateral-mode external cavity laser in this case.The reason is that the single-lobe in the far-field, in this case, is localizedat the same angle as the stripe mirror thus making the feedback much moreeffective than in the case where the mirror is located at approximately theangle opposite the optical axis. The field is then fed back to the positionof the near-field from which it was emitted, making way for a very efficientfeedback laser. One can realize a feedback setup resulting in the describedmode of operation by inserting an additional lens in the external cavity [94].Related to this discussion is the case of a phase-conjugate feedback. In thiscase the field fed back to the laser is (for a unfiltered feedback) [30]

EPC(x, t) ∝ E∗(x, t− τext). (5.23)

There is no minus in front of the x; instead the field is the complex conju-gate of the emitted field. This is readily extended to the filtered case. Anexperimental example of phase-conjugate feedback may be found in [95].

5.7 Summary

For the AEC laser we extended the z-averaged field equation to include an ex-ternal cavity with a spatially filtered feedback and obtained good qualitativeagreement with time-averaged measurements; a single-lobed time-averagedfar-field was calculated for an appropriate position of the stripe mirror. Theeffect of the external mirror on the time-varying near-field was seen to bea prevention of the lateral movement of filaments which is common for afreely running BA laser. The field was, however, still varying in time. Theimproved order in the near-field led to a far-field where most of the intensitywas fluctuating around an angle off the laser axis, approximately opposite tostripe mirror.

The phenomenological description of the gain material using linear gainand the α-parameter has to ad hoc be supplemented with an wavenumber-dependent loss. This method was found in the literature on time-dependent

105

modeling of BA lasers, and it is necessary in order to avoid unphysical am-plification of field components with high spatial frequency. The phenomeno-logical model was seen to be able to capture the dynamic filamentation ofthe solitary BA laser giving zig-zag patterns.

106

Chapter 6

Modal expansion of lateralmodes

In chapter 3 we found a wide variety of stationary states. The modes mI

were definite-parity modes. They have conventional threshold pump ratesat which the field is zero in the absence of noise. For currents modestlyabove the threshold, the field of the regarded mode becomes finite and thenonlinear light-semiconductor interaction causes the field to change shape.Take Figure 3.3 where the near-field of 6I just above threshold (a) changes byan increased dip in the middle of the near-field when increasing the current(b). Similar behavior is seen for the near-field of 5I in Figure 3.4 (a) and (b).This behavior can be understood by regarding the nonlinear mode mI as alinear gain guided mode perturbed by the nonlinearity of the gain material aswe shall show in the following. We do this by introducing a mode-expansiontechnique to calculate nonlinear lateral field distributions. The lateral field isexpanded in linear gain guided modes. As opposed to Chapter 3 we includecarrier diffusion and regard its effect on the field at low currents. To makeour results easy to interpret we neglect the spreading of current.

6.1 Expansions of the lateral field and the

carrier density

We denote the nonlinear stationary states calculated in this chapter Em,where m is the mode number. Let Em(x) and Φm(x) be related by ascaling factor ξm, that is, Em = ξmΦm, and let Φm be normalized such

107

that∫∞−∞ Φ2

mdx = 1. The lateral modes Φm are expanded in a completebiorthonormal set, i.e.

Φm =∞∑

q=1

b qmψq, (6.1)

where the modes ψq are eigenfunctions of the operator H0:

H0ψq = β20, qψq, (6.2)

and H0 is defined as

H0 =d2

dx2+ k2

tr(x), (6.3)

and β20, q are the eigenvalues corresponding to ψq. The wavenumber ktr is

defined through

k2tr(x) =

k2

r for |x| < x0

k2r − 2kr

∂k∂NNr for |x| > x0.

(6.4)

Recall that Nr = N0 + αi/(Γa). The wavenumber ktr is to describe the BAlaser at transparency provided that no spreading of current takes place. Themode Φm is to be understood through its near-field |Φm|2 having m dominantpeaks implying bmm > bq 6=m

m . Now, the full field problem is written as

(H0 +H ′)Φm = β2mΦm, (6.5)

where the operator H ′ is introduced as a perturbation that will modify theunperturbed modes ψq and their corresponding eigenvalues β2

0, q. H′ is defined

as

H ′ =

2kr

[1vg

(ωs − ωr) + ∂k∂N

(N −Nr)]

for |x| < x0

2kr1vg

(ωs − ωr) for |x| > x0.(6.6)

Inserting the expansion of Eq. (6.1) in (6.5), multiplying by ψk on bothsides, and then integrating gives us

∑

q

[H ′kq + δkq(β

20, q − β2

m)]bqm = 0. (6.7)

In Eq. (6.7) δ is Kronecker’s delta and

H ′kq =

∫ ∞

−∞ψkH

′ψqdx. (6.8)

108

Further, we have used that∫∞−∞ ψkψqdx = δkq. Also, as in Eq. (3.31) βm =

kr + αm

2. This constant value of βm for all m may be inserted in Eq. (6.7).

For clarity we rename it γ1 = kr + αm

2and Eq. (6.7) becomes

∑

q

[H ′kq + δkq(β

20, q − γ2

1)]bqm ≡

∑

q

Akqbqm = 0. (6.9)

We express the carrier density N as a Fourier series on |x| ≤ x0, i.e.

N = N +

∞∑

u=1

(∆Nue

jKux + c.c.). (6.10)

Further, we impose periodic boundary conditions:

Kux0 = uπ. (6.11)

Inserting (6.10) in Eq. (3.43) and then averaging from −x0 to x0 yields

−N+JτR =1

Psat

[(N −N0)|E0, m|2 +

1

2x0

∫ x0

−x0

|Em(x)|2∑

u

(∆NuejKux + c.c.)dx

],

(6.12)while projecting onto exp(∓jKix) gives

ηi∆Ni =1

Psat

[(N −N0)v(Ki) +

∑

u6=i

∆Nuv(Ki −Ku) +∑

u

∆N∗uv(Ki +Ku)

],

(6.13)and

ηi∆Ni =1

Psat

[(N −N0)v(−Ki) +

∑

u6=i

∆Nuv(−Ki −Ku) +∑

u

∆N∗uv(−Ki +Ku)

].

(6.14)We have used

|E0,m|2 =1

2x0

∫ x0

−x0

|Em|2dx, (6.15)

ηi = −DK2i τR − 1 − |E0, m|2

Psat

, (6.16)

and

v(Ki) =1

2x0

∫ x0

−x0

|Em(x)|2 exp(−jKix)dx. (6.17)

109

Eq. (6.12) scales the intensity |Em(x)|2 for given J , N , and coefficients ∆Nu,giving the proper ξ2

m. The properly scaled field can then be used in the linearsystem of Eq. (6.13) and (6.14) to solve for ∆Nu.

To summarize this section: Eq. (6.9) and Eqs. (6.12) together with(6.13) and (6.14) constitute the nonlinear system of equations that togetherwith the modes ψq describe a BA laser where lateral spreading of carriers isneglected. They must be solved iteratively.

6.2 Linear gain guided modes

We must find solutions to the unperturbed eigenvalue problem (6.2) to obtainthe modes for the expansion in Eq. (6.1). The lateral current profile isapproximated to be constant within x < |x0| and not to spread into x >|x0|while demanding ψq(x) and ∂xψq(x) to be continuous at ±x0. Choosingsuch simple boundary conditions will make our results easier to interpret,which will be evident in our discussion.The profile in Eq. (6.4) describesa gain guided lateral structure. Assuming solutions of region 1 (|x| ≤ x0)and region 2 (|x| > x0) of the form Cie

jkix + Die−jkix with i = 1, 2 and

substituting into Eq. (6.2), one may obtain

k21,q + β2

0, q = k2r (6.18)

k22,q + β2

0, q = k2r − 2kr

∂k

∂NNr = k2

r + kr∆α(α− j) (6.19)

and in addition the relation

k22, q − k2

1,q = kr∆α(α− j), (6.20)

where ∆α = ΓaNr = αi + ΓaN0 is the difference between the losses in theunpumped region 2 and the pumped region region 1. Since the unperturbedsystem is linear and has inversion symmetry the solutions must have definiteparity. The symmetric solutions are (integer q odd)

ψq(x) =

Bq exp(−jk2, qx) forx < −x0

Aq cos(k1, qx) for |x| < x0

Bq exp(jk2, qx) forx > x0.(6.21)

The asymmetric solutions are (integer q even)

ψq(x) =

−Bq exp(−jk2, qx) forx < −x0

Aq sin(k1, qx) for |x| < x0

Bq exp(jk2, qx) forx > x0.(6.22)

110

The boundary conditions at x = ±x0 imply the transcendental equations

−k1, q tan(k1, qx0) = jk2, q , q odd, (6.23)

andk1, q cot(k1,qx0) = jk2, q , q even. (6.24)

Eqs. (6.23) and (6.24) together with Eq. (6.20) must be solved numericallyfor a given set of parameters. In general k1, q, k2, q are complex. The spatialFourier frequency Ki associated with the carrier density has a magnitudethat is very close to 2Re(k1, i). The imaginary part of k1, q implies a slowlateral variation in ψq(x) on |x| < x0 in addition to the fast variation causedby the spatial frequency Re(k1, q). In particular a dip may be seen aroundx = 0 in |ψq(x)|2 for q ≥ 3. This dip is also found using BPM [31]. Finally,with Eq. (6.18) we find the unperturbed complex propagation constants β0, q.

-80

-60

-40

-20

0

3.4995 3.4996 3.4997 3.4998 3.4999 3.5

Im(β

0,q)

(m

-1)

Re(β0,q)/k0

Figure 6.1: Calculated real- and imaginary parts of the unperturbed prop-agation constants β0,q for q = 1 through q = 28. The lowest order, q = 1,corresponds to the point with the smallest negative imaginary part in theupper right corner. Higher orders have successively higher loss.

For q going from 1 to 28 we show the real and imaginary parts of β0,q

in Fig. 6.1. β0, 1 is located in the upper right corner and the sequence endsβ0, 28 in the lower left corner. The sequence illustrates how mode 1 has thelowest loss and that the loss increases monotonously with increasing modenumber as we also saw in Chapter 3. The increasing loss is due to growingenergy flow in outward directions for increasingly high mode order.

111

6.3 Results of mode expansion

The numerical procedure of obtaining stationary solutions is described inAppendix F. It returns the expansion coefficients. The expansion represen-tation of the field allows one to interpret nonlinear perturbations. That is,which coefficients bq 6=m

m are significant.

6.3.1 Comparison of modal expansion with scattering-potential method

In order to compare the modal-expansion method of this chapter with themethod of finding mI-modes in Chapter 3 we first set the diffusion coeffi-cient equal to zero in the former method and also set the current spreadingdistance d of the latter method equal to zero. Of course, we expect the twomethods to agree well; at least for pump rates where Φm can be regarded asa perturbed ψm. Near-fields for mode 7 for J = 1.015J0 obtained by the twodifferent methods are shown together in Fig. 6.2. The two methods showgood agreement.

0

2

4

6

8

10

-150 -100 -50 0 50 100 150

Nea

r-fie

ld [A

.U]

x [µm]

Mode expansion (no diffusion)Scattering potential (no current spreading)

Figure 6.2: Near-field of mode 7 obtained by method of present chapter(Modal expansion) with D = 0 and by the method of Chapter 3 (Scatteringpotential) with d = 0. The pump rate is J = 1.015J0. The two methodsagree well.

112

6.3.2 Perturbation interpretation of low-current non-linear modes

To be able to understand the effects of spatial hole burning in more detail,we consider the case of infinite absorption outside the pumped region, i.e. for∆α going to infinity, as a reference. For this special case, the unperturbedmodes and their eigenvalues may be found analytically [56]. For |x| < x0 theunperturbed modes ψq are given as

ψinfq (x) =

1√x0

cos(kinf1, qx) for odd q

1√x0sin(kinf

1, qx) for even q.(6.25)

where2x0k

inf1, q = qπ, (6.26)

and q is a positive integer. For |x| > x0, ψinfq = 0 for all q. The modes

in (6.25) are thus truncated sinusoidals. The corresponding eigenvalues aregiven as

(β20, q)inf ' k2

r −q2π2

4x20

. (6.27)

For a given q, kinfq is real and has half the magnitude of the spatial frequency

associated with ∆Nq, that is Kq. Then kinf1, q lies very close to Re(k1, q), which

is much greater than Im(k1, q), and similarly (β20, q)inf lies close to Re(β2

0, q),which is much greater than Im(β2

0, q). Thus kinf1,q and (β2

0, q)inf serve as usefulreferences for interpretation in the following.

So far, we have regarded the field Φm as an expansion with one dom-inating unperturbed mode ψm. This allows us, within restrictions, to useperturbation theory, using ψinf

q , to interpret the solutions, which we haveobtained by solving the complete nonlinear system of equations.

We wish to identify the role of individual Fourier components ∆Nu. Forthis purpose we define M(m, u, q):

∆NuM(m, u, q) =

∫ x0

−x0

ψ infm (∆Nu exp(jKux) + c.c.)ψ inf

q dx, (6.28)

with the matrix elements

M(m, u, q) = δ(m−q−2u),0 + (−1)m+1 + δ(m−q+2u),0. (6.29)

Note that with inversion symmetry, we have taken ∆Nu to be real.

113

To understand the results of the nonlinear calculations, we turn to per-turbation theory. We settle for first order, which gives expansion coefficients

bmm = 1 (6.30)

and for q 6= m

b qm =

H ′qm

β20, m − β2

0, q

(6.31)

Again, the approximation of infinite ∆α offers us insight. Using (ψq)inf and(β0, q)inf in Eq. (6.31) gives for q 6= m

(b qm)inf = 2kr

∂k

∂N

(4x2

0

π2

) ∑∞u=1 ∆NuM(m, u, q)

q2 −m2(6.32)

For a mode Φm, we find from actual computations of Em that the mostimportant Fourier components of the carrier density distribution at low cur-rents are in general ∆Nm and ∆N1. ∆Nm is caused by ψm as the main spatialfrequency of |ψm|2 lies close to Km = 2kinf

m . In conjunction with ∆Nm, wefurther identify the matrix element M(m,m, 3m), which enters Eq. (6.32).We note that this element involves ψ3m, which one could call the first higherBragg order of the grating component ∆Nm. In fact, in our calculations, wefind this Bragg component of the field to be non-zero, i.e. b3m

m 6= 0. Slowvariations in the near-field cause ∆N1 to be significant. Via the matrix el-ements M(m, 1, m ± 2), ∆N1 will enter Eq. (6.32) causing βm±2

m ψm±2 to besignificant in the expansion of Φm. Hence, we have used the approximationof infinite ∆α to identify the most important matrix elements for the 2 re-garded Fourier coefficients of the carrier density that are typically the mostsignificant.

As stated in the beginning of this chapter Figures 3.3 and 3.4 show howthe dip in the middle of the near-field increases as the current is increasedjust above threshold. Beating between bmmψm, and bm±2

m ψm±2 yields an in-creased slow variation of approximate spatial frequency K1. As the currentis increased and the intensity increases, |∆N1| is going to increase causing|bm±2

m | to rise yielding a larger dip and in turn a larger |∆N1| and so on. Forhigher currents more coefficients become significant.

114

-10

-8

-6

-4

-2

0

2

5 10 15 20 25 30 35 40

log 1

0(|b

7q |2 )

q

No diffusionDiffusion included

Figure 6.3: The absolute squares of the expansion coefficients bq7 are plottedas a function of q. Both the case with diffusion and the case with no diffusionincluded in the calculation are shown. Pump rate is J = 1.015J0. The effectof diffusion is to diminish the coefficients bq 6=m

m .

6.3.3 Identification of most important expansion termsand the influence of carrier diffusion

We perform a low-current investigation of modes 1 and 7. First mode 7. Theabsolute squares of the expansion coefficients of mode m = 7 at J = 1.015J0

are given as an example in Fig. 6.3. Both the cases without diffusion (D = 0)and with diffusion (D = 30 cm/s2). Other parameters are as in Chapter 3.Clearly, the coefficient b77 is close to unity at this current. As predicted by theinfinite loss approximation and perturbation theory, the largest coefficientsbq 6=77 are b57 and b97 joined by the q = 3m-coefficient b217 . Also the higher-

order Bragg order at q = 5m = 35 is visible. It is rather striking to see thetraces of a nonlinear Bragg grating, albeit a theoretical one. The effect ofcarrier diffusion is to diminish the expansion coefficients bq 6=7

7 slightly. Thecorresponding near-fields can be seen in Figure 6.4. Two things can be seenfrom the comparison. The near-field with diffusion has a smaller dip aroundx = 0 because of the inhibited b7±2

7 ; and the total power of the near-fieldobtined with diffusion is higher. This is due to a reduction in spatial holeburning, caused by the diffusion. It can be shown that a reduced |∆Nm| formode Φm reduces the nonlinear loss. We lastly investigate the influence ofcarrier diffusion on E1 at a low pump rate. As one could expect it is small.

115

0

2

4

6

8

10

12

-150 -100 -50 0 50 100 150

Nea

r-fie

ld [A

.U]

x [µm]

No diffusionDiffusion included

Figure 6.4: Near-field of mode E7 for cases of no diffusion respectively in-cluded diffusion when calculating the field at J = 1.015J0. The dip aroundx = 0 is smaller when diffusion is included. The total output power is higherwhen computed including the diffusion due the diffusion-induced suppressionof spatial hole burning.

The comparison of coefficients bq1 in Figure 6.5 tell that carrier diffusion is ofminor importance for the fundamental mode. The corresponding comparisonbetween near-fields is displayed in Figure 6.6 and the near-fields for cases withand without diffusion are very similar.

6.4 Summary

We have compared calculations of modes 1 and 7 including and excluding,respectively, carrier diffusion. We used a mode expanion technique. Clearly,mode 7 was more affected by diffusion. Nevertheless, the qualitative features,namely the dip in the middle and the identification of the largest coefficientsbq 6=mm for m = 7 as bm±2

m and b3mm , also held true when the diffusion was

included. Based on this we believe that there is good reason to believe thatthe stability properties found in Chapter 4 of modes mI with m ≥ 3, whichwere calculated in Chapter 3, are likely to hold true also when diffusion isincluded. This seems very certain for the fundamental mode m = 1 as it isonly affected to a very small degree by carrier diffusion at low pump rates.

116

-12

-10

-8

-6

-4

-2

0

2

2 4 6 8 10 12 14 16 18 20

log 1

0(|b

1q |2 )

q

No diffusionDiffusion included

Figure 6.5: The absolute squares of the expansion coefficients bq1 plotted asa function of q. Both the case with diffusion and the case with no diffusionincluded in the calculation are shown. Pump rate is J = 1.003J0. The effectof diffusion is seen to be small for the most significant coefficients bq 6=1

1 .

0

0.5

1

1.5

2

2.5

-150 -100 -50 0 50 100 150

Nea

r-fie

ld [A

.U]

x [µm]

No diffusionDiffusion included

Figure 6.6: Near-fields of mode E1 for J = 1.003J0 calculated without diffu-sion and with diffusion respectively. The difference is small.

117

118

Chapter 7

Summary

In this thesis we presented and discussed calculations regarding lateral modesof broad area lasers. We explored the actual mode structure, calculating indi-vidual stationary solutions. We then performed a small-signal analysis withthe emphasis on the stability properties of the lateral modes. The mode ofoperation of an existing asymmetric external-cavity set-up was studied in thestudied in the time-domain. Lastly stationary solutions were studied closeto their their threshold using a modal-expansion technique suited for inter-pretation via perturbation theory. Throughout the thesis we used equationsobtained through a mean field approximation implying averaging over thelongitudinal direction of the laser.

A BA laser pumped at a considerable current is in a fluctuating state.Most measured intensity distributions presented in the literature are due totime-averaged measurements. When calculating lateral, stationary modes ina BA laser it is therefore a misconception to expect that calculated station-ary solutions should reproduce such time-averaged measurements. However,from a theoretical point of view it is of interest to know about the stationarymode structure, both from the point of view of fundamental laser theory andperhaps also for designing devices. The lateral mode structure turned out tobe much richer than merely the “ordinary” lateral modes normally associ-ated with BA lasers. Several different types of modes constituted a structurewhere different mode-types were found to be interrelated. Most notably isthe existence of asymmetric modes, which exist in spite of the symmetric lat-eral structure. One category of asymmetric modes has a single-lobed far fieldthat resembles the far-field of the asymmetric external cavity laser that we

119

studied in Chapter 5. We saw another category of asymmetric modes whosenear-field compressed itself towards the lateral edge of the active region inthe manner of an accordion as the pump rate was increased. In total wepresented four different categories of modes whose tuning curves were seento be part of a systematic structure of merging curves.

Due to the spatial dependence of the lateral modes, we employed a Green’sfunction method to study the small signal properties of some of the calculatedstationary solutions. By means of this method one can determine the linearstability of calculated stationary solutions. We demonstrated methods todetermine both stability and bifurcations of the saddle-point type and of theHopf-type respectively. For the saddle-node stability we stated the functionσ which in order to be in accordance with bifurcation theory must changesign whenever a new branch emerges on the tuning curve along which oneis running. We saw that σ fulfilled this requirement. In addition its signdetermined the local stability of a given stationary state

The stability analysis gave some general results. Firstly, for a BA laserof width 200 µm, we find no stable modes except at very low currents (cur-rents below 1.002 times the threshold current of the laser). This theoreticalresult is fundamental because it tells why a BA laser driven at considerablecurrents operate in a fluctuating state: all modes are unstable. We have nottested the stability of all types of modes found in Chapter 4. Nevertheless,experiments and time-domain simulations tell us that hypothetical stable sta-tionary solutions never yield a steady-state; the trajectory in some face spacewill newer settle to the point of the hypothetical stable mode. Specifically forthe fundamental lateral mode, we found that when increasing the pump ratefrom threshold it becomes unstable (it Hopf-bifurcates) around 1.001 timesthe threshold current. The first higher order mode Hopf-bifurcates around1.0016 times the threshold current. These values are of course dependenton the utilized parameters. All standard gain guided modes of higher order,i.e. order 3 or greater, become unstable immediately above threshold or arealternatively “born” unstable. The asymmetric modes conceivably relatedto the AEC laser are locally stable but are globally unstable, i.e. they sufferfrom a Hopf-instability.

We demonstrated the calculation of linewidth for the fundamental modeand found a value of 8.0 GHz at a current just under the one at which themode Hopf-bifurcates. We mentioned that the Green’s function method canalso be used to calculate noise spectra for stable stationary solutions. We

120

leave the calculation of noise spectra for future work. Further, it was shownhow the change of the oscillation frequency due to a static change in currentcan be predicted correctly by the small-signal analysis. This shows the gen-eral strength of the Green’s function approach. The small-signal analysis canalso be generalized to devices with a built-in lateral index variation. Thiscould be useful in the hunt for a stable lateral mode in wide aperture lasers.

For direct comparisons between experiments and theory on BA lasers notrunning at the very low currents mentioned above, one probably has to turnto time-domain simulations. We solved our phenomenological mean fieldequations in the time-domain using the hopscotch method. We extended thefield equation to include a term describing a spatially filtered feedback inorder to describe the regarded asymmetric external cavity laser. The modelwas capable of giving a qualitative agreement with experiment: when thestripe mirror of the external cavity was placed, say, to the right of the laseraxis, a dominant lobe appeared to the left of the laser axis when perform-ing a time average. This, however, occurs after one has carefully placed thestripe mirror at a specific optimum position. Such an optimization of theposition of the stripe mirror is also necessary in an experiment. At this op-timum configuration of the asymmetric external cavity, the laser is still in atime-dependent state. However, in the near-field the lateral movement of fil-aments has been inhibited and the dynamics of the near-field resemble thoseof a laser array more than those of a solitary broad area laser. In agreementwith existing literature we found that the phenomenological description ofthe gain material urges that one ad hoc includes a wavenumber-dependentloss to avoid amplification of field components of high spatial frequency. Wesaw that phenomenological model could capture the lateral dynamics of thesolitary broad area laser yielding zig-zag patterns of filaments in the near-field.

Summing up in broad terms, an analysis of the stationary lateral modestructure of a BA laser showed a rich and systematically structured varietyof modes. All modes subject to a small-signal stability analysis proved to beunstable except at very low pump rates where the two lowest order modesmight be stable. The instability of all (viewed) modes at considerable pumprates is the small-signal answer to the question why the output of broadarea lasers is in general fluctuating. In the time-domain we showed thesefluctuations and showed that an asymmetric external cavity could dampen

121

the fluctuations but could not bring the laser into a stationary state.

122

Appendix A

Energy density and power

For a static field the energy density is given by

W = 2ε0nrng

(|E+|2eαmz + |E−|2e−αmz

)|φ(y)|2 . (A.1)

If we assume that E+ and E− are independent of z, the longitudinal averageis

W (x, y) = 2ε0nrng|φ(y)|2|E−(x)|2 1

l

∫ l

0

(r21e

αmz + e−αmz)dz

= 2ε0nrng|φ(y)|2K|Es(x)|2 (A.2)

where

K =(r1 + r2)(r1r2 − 1)

2r1r2ln(r1r2). (A.3)

We have here used that for z-independent E+ and E−, the boundary condi-tion E+ = r1E

− and the definition

Es(x) =1√2r1

(E+(x) + r1E−(x)) (A.4)

give the relations

Es(x) =√

2r1E− =

√2

r1E+ . (A.5)

The average energy in the active region (y = 0) is then

W (x, 0) = 2ε0nrng|φ(0)|2K|Es(x)|2 . (A.6)

123

The transverse confinement factor Γ is given by

Γ =

∫active layer

|φ(y)|2dy∫∞−∞ |φ(y)|2dy ' |φ(0)|2h (A.7)

where h is the active layer thickness, and where we have used that the dis-tribution φ(y) is normalized such that

∫∞−∞ |φ(y)|2dy = 1. We can therefore

replace |φ(0)|2 in (A.6) by Γ/ha.The output power per unit length at the right facet (i.e. the right facet

near field power) is

P2(x) = 2ε0nrc|E+|2eαml(1 − r22)

= 2ε0nrc|Es|21

2r2(1 − r2

2) . (A.8)

The left facet near field power is similarly

P1(x) = 2ε0nrc|E−|2(1 − r21)

= 2ε0nrc|Es|21

2r1(1 − r2

1) . (A.9)

The total output power per unit length is then

Ptot(x) = P1(x) + P2(x) = 2ε0nrcαmlK|Es(x)|2 . (A.10)

This is consistent with the fact that the total photon energy per unit laterallength decays with the rate vgαm, i.e.

Ptot(x) = vgαml

∫ ∞

−∞W (x, y)dy = 2ε0nrcαmlK|Es(x)|2 (A.11)

and then the total output power is given as

Pout =

∫ ∞

−∞Ptot(x)dx. (A.12)

124

Appendix B

Boundary conditions

In obtaining Eq. (4.31) we neglected dψ/dx at the interval boundaries

∂

∂xE(−A) = j

√κE(−A). (B.1)

In this appendix we omit the “WKB”-subscript on κ. Taking the differentialon both sides implies

∂

∂xδE(−A) = j

∂√κ

∂ωδωE(−A) + j

∂√κ

∂JδJE(−A) + j

√κδE(−A) (B.2)

or in the time domain where jδω is replaced by d/dt

∂

∂xδE =

∂√κ

∂ω

d

dtE + j

∂√κ

∂JδJE + j

√κδE (B.3)

We want the boundary conditions for δb = δE/E. The first derivative of δbis

d

dxδb = −δE

E

1

E

d

dxE +

1

E

d

dxδE (B.4)

Using (B.1) and (B.3) in (B.4) results in

d

dxδb =

∂√κ

∂ω

d

dtb+ j

∂√κ

∂JδJ, (B.5)

where db/dt = (1/E)dE/dt has been used. Hence,

d

dxψ =

d

dt

(Re(∂

√κ

∂ωb)

Im(∂√

κ

∂ωb)

)+

(−Im(∂

√κ

∂J)

Re(∂√

κ

∂J)

)δJ (B.6)

125

andd

dxψ = s

(Re(∂

√κ

∂ωb)

Im(∂√

κ

∂ωb)

)+

(−Im(∂

√κ

∂J)

Re(∂√

κ

∂J)

)δJ. (B.7)

Similarly for the right hand side x = A

d

dxδb = −∂

√κ

∂ω

d

dtb− j

∂√κ

∂JδJ, (B.8)

d

dxψ = −s

(Re(∂

√κ

∂ωb)

Im(∂√

κ

∂ωb)

)+

(Im(∂

√κ

∂J)

−Re(∂√

κ

∂J)

)δJ. (B.9)

We see from (B.7) and (B.9 ) that the local stability is not affected by theboundary conditions for a constant current (δJ = 0) since the local stabilityis associated with the system determinant D(s) at s = 0.

For the static frequency tuning, the terms including s equal zero. Theequation for static frequency tuning including the effect of boundary condi-tions is given as (cf. Eq. (4.93))

δω = δωB + kr

∂g

∂N

∫ A

−A

lims→0

sζ†2(x, s)

(α−1

)τRδJ

1 + |E(x)|2/Psat

dx, (B.10)

where

δωB = − lims→0

sζ†2(x, s)d

dxψ

∣∣∣∣A

−A

. (B.11)

After some derivations one obtains,

δωB = −σ−1

(y4)1(A)

[(d

dxψ(x)

)

1

(y3)3(x) +

(d

dxψ(x)

)

2

(y3)4(x)

]+

(y3)1(A)

[(d

dxψ(x)

)

1

(y4)3(x) +

(d

dxψ(x)

)

2

(y4)4(x)

]∣∣∣∣A

−A

.

(B.12)

Inserting from (B.7) and (B.9) gives

δωB = −σ−1

(Im

(∂√κ

∂J

)(y4)1(A) [(y3)3(A) + 1] − (y3)1(A)(y4)3(A)

−Re

(∂√κ

∂J

)(y4)1(A)(y3)4(A) − (y3)1(A) [(y4)4(A) + 1]

)δJ.

(B.13)

126

The derivative∂√κ

∂J= krτR

∂k

∂N

1√κ

(B.14)

is much smaller ( 10−26) than the various the various other entries. Theboundary conditions are therefore neglible on frequency tuning. This hasbeen confiremed numerically.

127

128

Appendix C

The stability parameter σ

Here we derive the asymptotic behavior of D(s) when s→ ∞ along the realaxis. For large positive s we can approximate the matrix M †

u defined in(4.37) by

M †u =

(0 I

−sMω 0

)(C.1)

The eigenvalues of this matrix are solutions to the characteristic equation

det(M †u − λ) = 0, (C.2)

which becomesλ4 + s′2 = 0 (C.3)

where s′ is the scaled parameter

s′ = 2kr

vg

s. (C.4)

The eigenvalues of M †u are then the four complex numbers λ1 = ej π

4

√s′,

λ2 = −ej π4

√s′, λ3 = e−j π

4

√s′, and λ4 = −e−j π

4

√s′. The eigenvector wi

corresponding to the eigenvector λi is

wTi = (s′λi,−λ3

i , s′λ2

i , (s′)2). (C.5)

From Eqs. (4.64)d

dxY †(x, s)wi = λiY

†wi. (C.6)

129

The solutions to (C.6) leads to

Y †(A, s)wi = eλi2A)wi (C.7)

We expand e1 and e2

e1 =

4∑

i=1

b1iwi (C.8)

and

e2 =4∑

i=1

b2iwi, (C.9)

which alternatively can be expressed

V b1 = e2 (C.10)

andV b2 = e4 (C.11)

where (b1)i = b1i and (b2)i = b1i for i = 1, .., 4. The matrix V is defined as

V =

(s′)2 (s′)2 (s′)2 (s′)2

s′λ1 s′λ2 s′λ3 s′λ4

s′λ21 s′λ2

2 s′λ23 s′λ2

4

−λ31 −λ3

2 −λ33 −λ3

4

. (C.12)

The determinant of V is related to the Vandermonde determinant. It isgiven by

detV = −(s′)4∏

i>j

(λi − λj). (C.13)

and inserting the eigenvalues λi

detV = 16(s′)7. (C.14)

From Eqs. (4.56), (4.57), (C.8), and (C.9) we get

q3(s) = Y †(A, s)e1 =∑

b1ieλi2Awi (C.15)

q4(s) = Y †(A, s)e2 =∑

b2ieλi2Awi. (C.16)

130

Using the expression for the system determinant D(s) in Eq. (4.61) thenyields

D(s) =∑

i,j

b1ib2je(λi+λj)2A((wi)3(wj)4 − (wi)4(wj)3) (C.17)

or inserting wi

D(s) = (s′)3∑

i,j

b1ib2je(λi+λj)2A(λ2

i − λ2j). (C.18)

By introducing the matrix B with matrix elements

(B)ij = (λ2i − λ2

j)e(λi+λj)2A (C.19)

the system determinant can be written

D(s) = (s′)3bT1Bb2. (C.20)

The squares of the eigenvalues λi are

λ21 = λ2

2 = js′ (C.21)

λ23 = λ2

4 = −js′. (C.22)

Then B may be written as

B =

(0 Bs

−BTs 0

), (C.23)

where Bs is the 2×2 submatrix

Bs = 2js′

(e2

√2s′A ej2

√2s′A

e−j2√

2s′A e−2√

2s′A

). (C.24)

The dominant factor is e2√

2s′A so for large s the system determinant becomes

D(s) = 2j(s′)4(b11b23 − b13b21)e2√

2s′A[1 + O(e−2√

2s′A)]. (C.25)

Solving Eq. (C.10) and (C.11) gives b1 and b2 from which the relevantcomponents are

b11 =−4(s′)5λ4

detV(C.26)

131

b13 =4(s′)5λ1

detV(C.27)

b21 =−4j(s′)5λ4

detV(C.28)

b23 =−4j(s′)5λ1

detV. (C.29)

These finally give the asymptotic form of D(s) for large s:

D(s) =1

4s′e2

√2s′A[1 + O(e−2

√2s′A)]. (C.30)

132

Appendix D

The Diffusion matrix D(x, s)

The diffusion matrix D(x, ω) defined by (4.82) originates from the Langevindriving term F (r, s) in (3.7), which describes the spontaneous emission noise.We will in this appendix derive an expression for D(x, s) based on the cor-relation relations for F (r, ω), where r = (x, y, z).

It has been shown by Henry [49] that F (r, ω) obeys the correlation rela-tions

〈F (r, ω)F (r′, ω′)〉 = 〈F ∗(r, ω)F ∗(r′, ω′)〉 = 0

〈F (r, ω)F ∗(r′, ω′)〉 = DF (r, ω)δ(r − r′)2πδ(ω − ω′) (D.1)

where

DF (r, ω) =2ω3~

c3ε0nrgmnsp . (D.2)

Here nr is the refractive index, gm the material gain, and nsp the spontaneousemission factor. They all depend on space and frequency.

From the definitions (3.14) and (3.23) of f±ω (x, z) and fω(x) it follows by

(D.1) that

〈fω(x)fω′(x′)〉 = 〈f ∗ω(x)f ∗

ω′(x′)〉 = 0

〈fω(x)f ∗ω′(x′)〉 = Df(x, ω)δ(x− x′)2πδ(ω − ω′) (D.3)

where

Df (x, ω) =1

2r1l2

∫ ∞

−∞

∫ l

0

DF (r, ω)|φ(y)|2(r2

1eαmz + e−αmz)

dydz . (D.4)

133

We will approximate Df(x, ω) by

Df (x, ω) ' 2ω3~

c3ε0lKnrgnsp, (D.5)

and g =∫∞−∞ gm|φ(y)|2dy is the modal gain. All parameters nr, g and nsp are

averaged over z. K is the Peterman factor (A.3).The relation analogous to (3.24) between f(x, t) and fω(x) leads to the

following relations between fω(x) and the components of f(x, s):

(f)1(x, s) = Ref(x, s) =1

2

(fωs+Ω(x)

Es(x)+f ∗

ωs−Ω(x)

E∗s (x)

)(D.6)

(f)2(x, s) = Imf(x, s) =1

2j

(fωs+Ω(x)

Es(x)− f ∗

ωs−Ω(x)

E∗s (x)

)(D.7)

where s = jΩ. From the correlation relations (D.3) for fω(x) we can finally

obtain correlation relations for the components of f(x, s) and thereby explicitexpressions for the elements of the diffusion matrix in (4.82). They read

(D)11 = (D)22 =1

4|Es|2(Df(x, ωs + Ω) +Df (x, ωs − Ω))

(D)12 = (D)∗21 =1

4j|Es|2(Df(x, ωs + Ω) −Df(x, ωs − Ω)) . (D.8)

In this paper we disregard the frequency dependence of Df , i.e. we will usethe approximation

D ' Df(x, ωs)

2|Es|2I . (D.9)

134

Appendix E

The hopscotch method

We use the hopscotch method to integrate Eq. (5.1) and (5.2).The x-axis is discretized and “grid” points p are equidistant and fixed.

When advancing one time increment ∆t, the points with even p and odd p,respectively, are treated differently. A hopscotch cycle consists of four parts.Firstly, field and carrier density at points with even p are advanced explicitlyin time. Secondly, field and carrier density at points with even p are advancedimplicitly in time. In the last two parts of the cycle the order of the two firstparts is changed. Thus in the third part, field and carrier density at pointswith odd p are advanced explicitly and in the fourth part field and carrierdensity at points with odd p are advanced implicitly. The implicit steps yieldnonlinear systems of difference equations which are solved through Newtoniteration. The Jacobian for this purpose is derived analytically.

First we show how to explicitly advance the field and carrier density astep in time, which is trivial. Thereafter we give the equations for the implicitstep. Since our regarded system of PDEs is nonlinear, the implicit part willinvolve Newton-iteration.

The equations for the field in (5.1) may be written

j + ε

2kr

∂2

∂x2E+

1

vg

∂

∂tE−1 + jα

2

∂g

∂N(N−N0)E+

(αi + αm

2

)E+j

ααi

2E = f(x, t)

(E.1)when including the small real number ε. Also, let us restate Eq. (5.2):

∂

∂tN(x, t) = J(x, t) +D

∂2

∂x2N − N(x, t)

τR− vggm(x, t)B|E(x, t)|2. (E.2)

135

We will find it useful to separate the field equation into real and imaginaryparts: The real part is given as

− ε

2kr

∂2

∂x2ER − 1

2kr

∂2

∂x2EI +

1

vg

∂

∂tER +

1

2Γa(N −N0)(αEI − ER)+

(αi + αm

2

)ER − ααi

2EI = fR(x, t),

(E.3)

while the imaginary part reads

− ε

2kr

∂2

∂x2EI +

1

2kr

∂2

∂x2ER +

1

vg

∂

∂tEI −

1

2Γa(N −N0)(αER + EI)+

(αi + αm

2

)EI +

ααi

2ER = fI(x, t).

(E.4)

We now describe such a hopscotch cycle applied to our system of PDEsin more detail.

The field at the position xp is explicitly advanced one time increment as

Eq+1p = −jA r(Eq

p−1 + Eqp+1)+[

1 + 2jA r + ∆tvg

1 + jα

2Γa(N q

p −N0) + ∆tvg

αi + αm

2+ ∆tvgj

ααi

2

]Eq

p+

∆tvgfqp ,

(E.5)

where r = ∆t/(∆x)2 and A = vg(1 + jε)/(2kr). The carrier density at theposition xp is explicitly advanced one time increment as

N q+1p = ∆t(Jq

p − vgaN0B|Eqp|2) +Dr(N q

p+1 +N qp−1)+

(1 − 2Dr − ∆t

τR− ∆tvgaB|Eq

p|2)N qp .

(E.6)

Next, we use Eqs. (E.3) and (E.4) to get expressions for an implicitadvancement of the field. Let A ′ = vg/(2kr) and A ′′ = vgε/(2kr). For thereal part of the field equation one obtains

Eq+1R,p − Eq

R,p =A ′r(Eq+1I,p+1 − 2Eq+1

I,p + Eq+1I,p−1) + A ′′r(Eq+1

R,p+1 − 2Eq+1R,p + Eq+1

R,p−1)−

vg

1

2Γa∆t(N q+1

p −N0)(αEq+1I,p − Eq+1

R,p )−(αi + αm

2

)∆tvgE

q+1R,p + ∆tvg

ααi

2Eq+1

I,p + ∆tvgfq+1R,p

(E.7)

136

Rearranging terms in (E.7) and defining the function T1(Eq+1R,p , E

q+1I,p , N

q+1p )

yields

0 = EqR,p − Eq+1

R,p

[1 − ∆tvg

1

2Γa(N q+1

p −N0) + ∆tvg

(αi + αm

2

)+ 2A ′′r

]

− Eq+1I,p

[2A ′r + α∆tvg

1

2Γa(N q+1

p −N0) − ∆tvg

ααi

2

]+ A ′′r(Eq+1

R,p+1 + Eq+1R,p−1)+

rA ′(Eq+1I,p+1 + Eq+1

I,p−1) + ∆tvgfq+1R,p ≡ T1(E

q+1R,p , E

q+1I,p , N

q+1p ).

(E.8)

Likewise for the imaginary part of the field equation

Eq+1I,p − Eq

I,p = −A ′r(Eq+1R,p+1 − 2Eq+1

R,p + Eq+1R,p−1) + A ′′r(Eq+1

I,p+1 − 2Eq+1I,p + Eq+1

I,p−1)+

∆tvg

1

2Γa(N q+1

p −N0)(αEq+1R,p + Eq+1

I,p )

− ∆tvg

(αi + αm

2

)Eq+1

I,p − ∆tvg

ααi

2Eq+1

R,p + ∆tvgfq+1I,p .

(E.9)

Also rearranging terms in (E.9) and defining the function T2(Eq+1R,p , E

q+1I,p , N

q+1p )

yields

0 = EqI,p − Eq+1

I,p

[1 − ∆tvg

1

2Γa(N q+1

p −N0) + ∆tvg

(αi + αm

2

)− 2A ′′r

]

+ Eq+1R,p

[2A ′r + α∆tvg

1

2Γa(N q+1

p −N0) − ∆tvg

ααi

2

]−

A ′r(Eq+1I,p+1 + Eq−1

I,p−1) + A ′′r(Eq+1R,p+1 + Eq−1

R,p−1) + ∆tGq+1I,p ≡M3(E

q+1R,p , E

q+1I,p , N

q+1p ).

(E.10)

The carrier density at point q is advanced implicitly as

N q+1p −N q

p = ∆tJq+1p +Dr(N q+1

p−1−2N q+1p +N q+1

p+1 )−∆t

τRN q+1

p −∆tvga(Nq+1p −N0)B|Ep+1

q |2.(E.11)

Now (E.11) is rearranged and the function T3(Eq+1R,p , E

q+1I,p , N

q+1p ) is defined

0 = N qp −N q+1

p (1 + 2Dr +∆t

τR+ ∆tvgaB|Ep+1

q |2) +Dr(N q+1p+1 +N q+1

p−1 )+

∆tJq+1p + ∆tN0vgaB|Ep+1

q |2 ≡ T3(Eq+1R,p , E

q+1I,p , N

q+1p ).

(E.12)

137

When advancing implicitly we must solve the nonlinear set of equations forthe three unknowns Eq+1

R,p , Eq+1I,p , and N q+1

p , i.e.

T1(Eq+1R,p , E

q+1I,p , N

q+1p ) = 0 (E.13)

T2(Eq+1R,p , E

q+1I,p , N

q+1p ) = 0 (E.14)

T3(Eq+1R,p , E

q+1I,p , N

q+1p ) = 0. (E.15)

The unknowns are computed through Newton iteration. For this purpose weneed the Jacobian whose nine elements are seen to be

∂T1

∂Eq+1R,p

= −[1 − ∆tvg

1

2Γa(N q+1

p −N0) + ∆tvg

(αi + αm

2

)+ 2A ′′r

]

(E.16)∂T1

∂Eq+1I,p

= −[2A ′r + α∆tvg

1

2Γa(N q+1

p −N0) − ∆tvg

ααi

2

](E.17)

∂T1

∂N q+1p

= ∆tvg

1

2Γa(Eq+1

R,p − αEq+1I,p ) + ∆tvg

∂f q+1R,p

∂N q+1p

(E.18)

∂T2

∂Eq+1R,p

= 2A ′r + α∆tvg

1

2Γa(N q+1

p −N0) − ∆tvg

ααi

2(E.19)

∂T2

∂Eq+1I,p

= −[1 − ∆tvg

1

2Γa(N q+1

p −N0) + ∆tvg

(αi + αm

2

)− 2A ′′r

]

(E.20)

∂T2

∂N q+1p

= ∆tvg

1

2Γa(αEq+1

R,p + Eq+1I,p ) + ∆tvg

∂f q+1I,p

∂N q+1p

(E.21)

∂T3

∂Eq+1R,p

= 2Eq+1R,p (N0 −N q+1

p )∆tvgaB (E.22)

∂T3

∂Eq+1I,p

= 2Eq+1I,p (N0 −N q+1

p )∆tvgaB (E.23)

∂T3

∂N q+1p

= −(1 + 2Dr +∆t

τR+ ∆tvgaB|Ep+1

q |2). (E.24)

One advantage of the hopscotch method is that after advancing the fieldand carrier density explicitly at points, say, even p, the only unknowns in theimplitic integration at a given odd p are Eq+1

R,p , Eq+1I,p , and N q+1

p .

138

The spatially filtering delay term in Eq. (5.18) is easily included. Thetime delay involves a stack, and the integral may either be obtained throughdirect integraton or by using Fast Fourier Transforms. We have done theformer.

139

140

Appendix F

Modal expansion -method ofsolution

Having established the model equations, we now describe the procedure ofcalculating a stationary solution, i.e. a point in the (ωs, N)-plane along withthe coefficients ∆Nu and the lateral field distribution Em(x) = ξmφm(x).Then Eqs. (6.12), (6.13), and (6.14) governing the carrier density and Eqs.(6.9) with (6.6) governing the field constitute the nonlinear set of algebraicequations to be solved iteratively.

Initially, we must choose the mode order m of the mode Φm that wewish to regard. The iteration-scheme is prepared (the zeroth iteration) by

setting all ∆N(0)u = 0 implying Φ

(0)m = ψm (the iteration number is denoted

in the parenthesis). Then only the mth equation in the system of Eq. (6.9)has to be solved for (ωs, N)(0). With the obtained (ωs, N)(0), Eq. (6.12) is

used to scale the field, i.e. to find ξ(0)m as |E(0)

m |2 = (ξ(0)m )2|φ(0)

m |2. The coreof the iteration-scheme may now begin by solving the linear system of Eq.(6.13) for the Fourier coefficients ∆N

(1)u using the scaled field. This is easily

done numerically [96]. The calculated coefficients ∆N(1)u are then inserted

into the full system of equations of Eq. (6.9). To find a solution of Eq.(6.9), (ωs, N)(1) must be found such that the determinant of the matrix bezero. A Newton-Raphson algorithm is used for this. We now have a singularmatrix, and the coefficients (bqm)(1) may be obtained using a singular value

decomposition [61][96]. This gives Φ(1)m by Eq. (6.1) and the scaling factor

ξ(1)m by means of Eq. (6.12). With the scaled field |E(1)

m |2, Eq. (6.13) is solved

for the Fourier coefficients ∆N(2)u and so on. Note that in (6.12) the field is

141

scaled using (ωs, N)(i) and ∆N(i−1)u . The iteration scheme and is continued

until (ωs, N), the coefficients ∆Nu, and |Em(x)|2 have converged. We findthat in general at least the first 3m + 2 modes ψq are needed in the modalexpansion for good convergence since field coefficients as high as b3m+2

m maybe non-negligible. In the calculations we use 100 modes.

142

Appendix G

Papers and presentations

Minjung Chi, Søren Blaaberg Jensen, Jean-Pierre Huignard and PaulMichael Petersen, “Two-wave mixing in a broad-area semiconductoramplifier”, submitted to Optics Express.

Paul M. Petersen, Eva Samsøe, Søren B. Jensen, and Peter Andersen,“Guiding of laser modes based on self-pumped four-wave mixing in asemiconductor amplifier”, Optics Express, Vol. 13, Issue 9, pp. 3340-3347.

Søren Blaaberg, Paul Michael Petersen, Bjarne Tromborg, “Structure oflateral modes of a broad area semiconductor laser, their stability andspectral properties”, in preperation.

Søren Blaaberg Jensen, Bjarne Tromborg, and Paul Michael Petersen.“Spatially nondegenerate four-wave mixing in a broad area semiconductorlaser: Modeling”, Northern Optics 2003, Poster presentation: June, 2003,Helsinki.

Søren Blaaberg Jensen, “A Model for Broad area lasers,” Biomedical Optics04, Talk: November 2004, Kgs. Lyngby.

Søren Blaaberg Jensen, “Spatially nondegenerate four-wave mixing in abroad area laser”, European Semiconductor Laser Workshop 2003,September 2003, Torino Italy.

143

144

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